Showing 1–47 of 47 results for author: Seif, A

Big cats: entanglement in 120 qubits and beyond

Authors: Ali Javadi-Abhari, Simon Martiel, Alireza Seif, Maika Takita, Ken X. Wei

Abstract: Entanglement is the quintessential quantum phenomenon and a key enabler of quantum algorithms. The ability to faithfully entangle many distinct particles is often used as a benchmark for the quality of hardware and control in a quantum computer. Greenberger-Horne-Zeilinger (GHZ) states, also known as Schrödinger cat states, are useful for this task. They are easy to verify, but difficult to prepar… ▽ More Entanglement is the quintessential quantum phenomenon and a key enabler of quantum algorithms. The ability to faithfully entangle many distinct particles is often used as a benchmark for the quality of hardware and control in a quantum computer. Greenberger-Horne-Zeilinger (GHZ) states, also known as Schrödinger cat states, are useful for this task. They are easy to verify, but difficult to prepare due to their high sensitivity to noise. In this Letter we report on the largest GHZ state prepared to date consisting of 120 superconducting qubits. We do this via a combination of optimized compilation, low-overhead error detection and temporary uncomputation. We use an automated compiler to maximize error-detection in state preparation circuits subject to arbitrary qubit connectivity constraints and variations in error rates. We measure a GHZ fidelity of 0.56(3) with a post-selection rate of 28%. We certify the fidelity of our GHZ states using multiple methods and show that they are all equivalent, albeit with different practical considerations. △ Less

Submitted 10 October, 2025; originally announced October 2025.

Transversal gates for probabilistic implementation of multi-qubit Pauli rotations

Authors: Nobuyuki Yoshioka, Alireza Seif, Andrew Cross, Ali Javadi-Abhari

Abstract: We introduce a general framework for weak transversal gates -- probabilistic implementation of logical unitaries realized by local physical unitaries -- and propose a novel partially fault-tolerant quantum computing architecture that surpasses the standard Clifford+T architecture on workloads with million-scale Clifford+T gate counts. First, we prove the existence of weak transversal gates on the… ▽ More We introduce a general framework for weak transversal gates -- probabilistic implementation of logical unitaries realized by local physical unitaries -- and propose a novel partially fault-tolerant quantum computing architecture that surpasses the standard Clifford+T architecture on workloads with million-scale Clifford+T gate counts. First, we prove the existence of weak transversal gates on the class of Calderbank-Shor-Steane codes, covering high-rate qLDPC and topological codes such as surface code or color codes, and present an efficient algorithm to determine the physical multi-qubit Pauli rotations required for the desired logical rotation. Second, we propose a partially fault-tolerant Clifford+$φ$ architecture that performs in-place Pauli rotations via a repeat-until-success strategy; phenomenological simulations indicate that a rotation of 0.003 attains logical error of $9.5\times10^{-5}$ on a surface code with $d=7$ at physical error rate of $10^{-4}$, while avoiding the spacetime overheads of magic state factories, small angle synthesis, and routing. Finally, we perform resource estimation on surface and gross codes for a Trotter-like circuit with $N=108$ logical qubits to show that the Clifford+$φ$ architecture outperforms the conventional Clifford+T approach by a factor of tens to a hundred in runtime due to natural rotation-gate parallelism. This work open a novel paradigm for realizing logical operations beyond the constraints of conventional design. △ Less

Submitted 9 October, 2025; originally announced October 2025.

Comments: 37 pages, 11 figures

Error mitigation of shot-to-shot fluctuations in analog quantum simulators

Authors: Thomas Steckmann, De Luo, Yu-Xin Wang, Sean R. Muleady, Alireza Seif, Christopher Monroe, Michael J. Gullans, Alexey V. Gorshkov, Or Katz, Alexander Schuckert

Abstract: Analog quantum simulators have provided key insights into quantum many-body dynamics. However, in such systems, both coherent and incoherent errors limit their scalability, hindering simulations in regimes that challenge classical simulations. In this work, we introduce an error mitigation technique that addresses and effectively suppresses a key source of error in leading simulator platforms: sho… ▽ More Analog quantum simulators have provided key insights into quantum many-body dynamics. However, in such systems, both coherent and incoherent errors limit their scalability, hindering simulations in regimes that challenge classical simulations. In this work, we introduce an error mitigation technique that addresses and effectively suppresses a key source of error in leading simulator platforms: shot-to-shot fluctuations in the parameters for the Hamiltonian governing the system dynamics. We rigorously prove that amplifying this shot-to-shot noise and extrapolating to the zero-noise limit recovers noiseless results for realistic noise distributions. Experimentally, we demonstrate this technique on a 27-ion trapped-ion quantum simulator, extending the two-qubit exchange oscillation lifetime threefold. Numerically, we predict a significant enhancement in the effective many-body coherence time for Rydberg atom arrays under realistic conditions. Our scheme provides a possible route towards extending the effective coherence time in analog quantum experiments, enabling deeper explorations of quantum many-body dynamics. △ Less

Submitted 19 June, 2025; originally announced June 2025.

Enhancing quantum noise characterization via extra energy levels

Authors: Senrui Chen, Akel Hashim, Noah Goss, Alireza Seif, Irfan Siddiqi, Liang Jiang

Abstract: Noise is a major challenge for building practical quantum computing systems. Precise characterization of quantum noise is crucial for developing effective error mitigation and correction schemes. However, state preparation and measurement (SPAM) errors on many current platforms can introduce large ambiguity into conventional noise characterization methods. In this work, we propose a scheme for enh… ▽ More Noise is a major challenge for building practical quantum computing systems. Precise characterization of quantum noise is crucial for developing effective error mitigation and correction schemes. However, state preparation and measurement (SPAM) errors on many current platforms can introduce large ambiguity into conventional noise characterization methods. In this work, we propose a scheme for enhancing quantum noise characterization using additional energy levels. We first develop a comprehensive theory on the identifiability of n-qudit SPAM noise given high-quality single-qudit control, showing the existence of gauge freedoms which can be completely described using subsystem depolarizing maps. We then show how to use these extra energy levels to reduce the gauge ambiguity in characterizing both SPAM and gate noise in the qubit subspace. We experimentally implement these ideas on a superconducting quantum computing device and demonstrate a qutrit-enabled enhancement in noise characterization precision. △ Less

Submitted 13 July, 2025; v1 submitted 10 June, 2025; originally announced June 2025.

Comments: 6 + 4 pages, 4 figures, comments welcome. v2: affiliation and acknowledgment updated

Disambiguating Pauli noise in quantum computers

Authors: Edward H. Chen, Senrui Chen, Laurin E. Fischer, Andrew Eddins, Luke C. G. Govia, Brad Mitchell, Andre He, Youngseok Kim, Liang Jiang, Alireza Seif

Abstract: To successfully perform quantum computations, it is often necessary to first accurately characterize the noise in the underlying hardware. However, it is well known that fundamental limitations prevent the unique identification of the noise. This raises the question of whether these limitations impact the ability to predict noisy dynamics and mitigate errors. Here, we show, both theoretically and… ▽ More To successfully perform quantum computations, it is often necessary to first accurately characterize the noise in the underlying hardware. However, it is well known that fundamental limitations prevent the unique identification of the noise. This raises the question of whether these limitations impact the ability to predict noisy dynamics and mitigate errors. Here, we show, both theoretically and experimentally, that when learnable parameters are self-consistently characterized, the unlearnable (gauge) degrees of freedom do not impact predictions of noisy dynamics or error mitigation. We use the recently introduced framework of gate set Pauli noise learning to efficiently and self-consistently characterize and mitigate noise of a complete gate set, including state preparation, measurements, single-qubit gates and multi-qubit entangling Clifford gates. We validate our approach through experiments with up to 92 qubits and show that while the gauge choice does not affect error-mitigated observable values, optimizing it reduces sampling overhead. Our findings address an outstanding issue involving the ambiguities in characterizing and mitigating quantum noise. △ Less

Submitted 28 May, 2025; originally announced May 2025.

Comments: 28 pages, 13 figures, 3 tables

Efficient quantum tomography of a polynomial subspace

Authors: Yat Wong, Ming Yuan, Kevin He, Srivatsan Chakram, Alireza Seif, David I. Schuster, Liang Jiang

Abstract: Quantum tomography is crucial for characterizing the quantum states of multipartite systems, but its practicality is often limited by the exponentially large dimension of the Hilbert space. Most existing approaches, such as compressed sensing and tensor network-based tomography, impose structural constraints on the state to enable more resource-efficient characterization. However, not all physical… ▽ More Quantum tomography is crucial for characterizing the quantum states of multipartite systems, but its practicality is often limited by the exponentially large dimension of the Hilbert space. Most existing approaches, such as compressed sensing and tensor network-based tomography, impose structural constraints on the state to enable more resource-efficient characterization. However, not all physical states can be well-approximated with highly structured states. Here, we develop a partial quantum tomography method based on direct fidelity estimation (DFE) that focuses on a neighborhood subspace -- the subspace spanned by states physically close to a given target state. Using this generalized DFE method, we estimate elements of the density operator within this subspace in a self-verifying manner. We investigate the efficiency of this approach under different sets of available measurements for various states and find that the set of available measurements significantly impacts the cost of DFE. For example, we show that Pauli measurements alone are insufficient for performing efficient DFE on all product states, whereas the full set of product measurements is sufficient. This method can be applied in many situations, including characterizing quantum systems with confined dynamics and verifying preparations of quantum states and processes. △ Less

Submitted 28 February, 2025; originally announced March 2025.

Efficient Lindblad synthesis for noise model construction

Authors: Moein Malekakhlagh, Alireza Seif, Daniel Puzzuoli, Luke C. G. Govia, Ewout van den Berg

Abstract: Effective noise models are essential for analyzing and understanding the dynamics of quantum systems, particularly in applications like quantum error mitigation and correction. However, even when noise processes are well-characterized in isolation, the effective noise channels impacting target quantum operations can differ significantly, as different gates experience noise in distinct ways. Here,… ▽ More Effective noise models are essential for analyzing and understanding the dynamics of quantum systems, particularly in applications like quantum error mitigation and correction. However, even when noise processes are well-characterized in isolation, the effective noise channels impacting target quantum operations can differ significantly, as different gates experience noise in distinct ways. Here, we present a noise model construction method that builds an effective model from a Lindbladian description of the physical noise processes acting simultaneously to the desired gate operation. It employs the Magnus expansion and Dyson series, and can be utilized for both low-order symbolic and high-order numerical approximations of the noise channel of a multi-qubit quantum gate. We envision multiple use cases of our noise construction method such as (i) computing the corresponding noise channel from a learned Lindbladian, and (ii) generating the noise channel starting with physically motivated Lindbladians for a given hardware architecture. In doing so, we close the gap between physical Lindbladians and operational level noise model parameters. We demonstrate a strong agreement between our symbolic noise construction and full numerical Lindblad simulations for various two-qubit gates, in isolation and in three- and four-qubit scenarios, for a variety of physically motivated noise sources. Our symbolic construction provides a useful breakdown of how noise model parameters depend on the underlying physical noise parameters, which gives qualitative insight into the structure of errors. For instance, our theory provides insight into the interplay of Lindblad noise with the intended gate operations, and can predict how local Lindblad noise can effectively spread into multi-qubit error. △ Less

Submitted 5 February, 2025; originally announced February 2025.

Comments: 24 pages, 9 figures, 5 tables

Quantum-Centric Algorithm for Sample-Based Krylov Diagonalization

Authors: Jeffery Yu, Javier Robledo Moreno, Joseph T. Iosue, Luke Bertels, Daniel Claudino, Bryce Fuller, Peter Groszkowski, Travis S. Humble, Petar Jurcevic, William Kirby, Thomas A. Maier, Mario Motta, Bibek Pokharel, Alireza Seif, Amir Shehata, Kevin J. Sung, Minh C. Tran, Vinay Tripathi, Antonio Mezzacapo, Kunal Sharma

Abstract: Approximating the ground state of many-body systems is a key computational bottleneck underlying important applications in physics and chemistry. The most widely known quantum algorithm for ground state approximation, quantum phase estimation, is out of reach of current quantum processors due to its high circuit-depths. Subspace-based quantum diagonalization methods offer a viable alternative for… ▽ More Approximating the ground state of many-body systems is a key computational bottleneck underlying important applications in physics and chemistry. The most widely known quantum algorithm for ground state approximation, quantum phase estimation, is out of reach of current quantum processors due to its high circuit-depths. Subspace-based quantum diagonalization methods offer a viable alternative for pre- and early-fault-tolerant quantum computers. Here, we introduce a quantum diagonalization algorithm which combines two key ideas on quantum subspaces: a classical diagonalization based on quantum samples, and subspaces constructed with quantum Krylov states. We prove that our algorithm converges in polynomial time under the working assumptions of Krylov quantum diagonalization and sparseness of the ground state. We then demonstrate the scalability of our approach by performing the largest ground-state quantum simulation of impurity models using a Heron quantum processors and the Frontier supercomputer. We consider both the single-impurity Anderson model with 41 bath sites, and a system with 4 impurities and 7 bath sites per impurity. Our results are in excellent agreement with Density Matrix Renormalization Group calculations. △ Less

Submitted 17 September, 2025; v1 submitted 16 January, 2025; originally announced January 2025.

Comments: 22 pages, 6 figures

Randomized benchmarking with non-Markovian noise and realistic finite-time gates

Authors: Antoine Brillant, Peter Groszkowski, Alireza Seif, Jens Koch, Aashish Clerk

Abstract: We analyze the impact of non-Markovian classical noise on single-qubit randomized benchmarking experiments, in a manner that explicitly models the realization of each gate via realistic finite-duration pulses. Our new framework exploits the random nature of each gate sequence to derive expressions for the full survival probability decay curve which are non-perturbative in the noise strength. In th… ▽ More We analyze the impact of non-Markovian classical noise on single-qubit randomized benchmarking experiments, in a manner that explicitly models the realization of each gate via realistic finite-duration pulses. Our new framework exploits the random nature of each gate sequence to derive expressions for the full survival probability decay curve which are non-perturbative in the noise strength. In the presence of non-Markovian noise, our approach shows that the decay curve can exhibit a strong dependence on the implementation method, with regimes of both exponential and power law decays. We discuss how these effects can complicate the interpretation of a randomized-benchmarking experiment, but also how to leverage them to probe non-Markovianty. △ Less

Submitted 25 February, 2025; v1 submitted 10 January, 2025; originally announced January 2025.

Exponential entanglement advantage in sensing correlated noise

Authors: Yu-Xin Wang, Jacob Bringewatt, Alireza Seif, Anthony J. Brady, Changhun Oh, Alexey V. Gorshkov

Abstract: In this work, we propose a new form of exponential quantum advantage in the context of sensing correlated noise. Specifically, we focus on the problem of estimating parameters associated with Lindblad dephasing dynamics, and show that entanglement can lead to an exponential enhancement in the sensitivity (as quantified via quantum Fisher information of the sensor state) for estimating a small para… ▽ More In this work, we propose a new form of exponential quantum advantage in the context of sensing correlated noise. Specifically, we focus on the problem of estimating parameters associated with Lindblad dephasing dynamics, and show that entanglement can lead to an exponential enhancement in the sensitivity (as quantified via quantum Fisher information of the sensor state) for estimating a small parameter characterizing the deviation of system Lindbladians from a class of maximally correlated dephasing dynamics. This result stands in stark contrast with previously studied scenarios of sensing uncorrelated dephasing noise, where one can prove that entanglement does not lead to an advantage in the signal-to-noise ratio. Our work thus opens a novel pathway towards achieving entanglement-based sensing advantage, which may find applications in characterizing decoherence dynamics of near-term quantum devices. Further, our approach provides a potential quantum-enhanced probe of many-body correlated phases by measuring noise generated by a sensing target. We also discuss realization of our protocol using near-term quantum hardware. △ Less

Submitted 8 October, 2024; originally announced October 2024.

Comments: 7+2 pages, 1 figure

Bounding the systematic error in quantum error mitigation due to model violation

Authors: L. C. G. Govia, S. Majumder, S. V. Barron, B. Mitchell, A. Seif, Y. Kim, C. J. Wood, E. J. Pritchett, S. T. Merkel, D. C. McKay

Abstract: Quantum error mitigation is a promising route to achieving quantum utility, and potentially quantum advantage in the near-term. Many state-of-the-art error mitigation schemes use knowledge of the errors in the quantum processor, which opens the question to what extent inaccuracy in the error model impacts the performance of error mitigation. In this work, we develop a methodology to efficiently co… ▽ More Quantum error mitigation is a promising route to achieving quantum utility, and potentially quantum advantage in the near-term. Many state-of-the-art error mitigation schemes use knowledge of the errors in the quantum processor, which opens the question to what extent inaccuracy in the error model impacts the performance of error mitigation. In this work, we develop a methodology to efficiently compute upper bounds on the impact of error-model inaccuracy in error mitigation. Our protocols require no additional experiments, and instead rely on comparisons between the error model and the error-learning data from which the model is generated. We demonstrate the efficacy of our methodology by deploying it on an IBM Quantum superconducting qubit quantum processor, and through numerical simulation of standard error models. We show that our estimated upper bounds are typically close to the worst observed performance of error mitigation on random circuits. Our methodology can also be understood as an operationally meaningful metric to assess the quality of error models, and we further extend our methodology to allow for comparison between error models. Finally, contrary to what one might expect we show that observable error in noisy layered circuits of sufficient depth is not always maximized by a Clifford circuit, which may be of independent interest. △ Less

Submitted 20 August, 2024; originally announced August 2024.

Comments: 22 pages including references and appendices, 9 figures

Entanglement-enhanced learning of quantum processes at scale

Authors: Alireza Seif, Senrui Chen, Swarnadeep Majumder, Haoran Liao, Derek S. Wang, Moein Malekakhlagh, Ali Javadi-Abhari, Liang Jiang, Zlatko K. Minev

Abstract: Learning unknown processes affecting a quantum system reveals underlying physical mechanisms and enables suppression, mitigation, and correction of unwanted effects. Describing a general quantum process requires an exponentially large number of parameters. Measuring these parameters, when they are encoded in incompatible observables, is constrained by the uncertainty principle and requires exponen… ▽ More Learning unknown processes affecting a quantum system reveals underlying physical mechanisms and enables suppression, mitigation, and correction of unwanted effects. Describing a general quantum process requires an exponentially large number of parameters. Measuring these parameters, when they are encoded in incompatible observables, is constrained by the uncertainty principle and requires exponentially many measurements. However, for Pauli channels, having access to an ideal quantum memory and entangling operations allows encoding parameters in commuting observables, thereby exponentially reducing measurement complexity. In practice, though, quantum memory and entangling operations are always noisy and introduce errors, making the advantage of using noisy quantum memory unclear. To address these challenges we introduce error-mitigated entanglement-enhanced learning and show, both theoretically and experimentally, that even with noise, there is a separation in efficiency between learning Pauli channels with and without entanglement with noisy quantum memory. We demonstrate our protocol's efficacy in examples including hypothesis testing with up to 64 qubits and learning inherent noise processes in a layer of parallel gates using up to 16 qubits on a superconducting quantum processor. Our protocol provides accurate and practical information about the process, with an overhead factor of $1.33 \pm 0.05$ per qubit, much smaller than the fundamental lower bound of 2 without entanglement with quantum memory. Our study demonstrates that entanglement with auxiliary noisy quantum memory combined with error mitigation considerably enhances the learning of quantum processes. △ Less

Submitted 6 August, 2024; originally announced August 2024.

Signatures of Quantum Phase Transitions in Driven Dissipative Spin Chains

Authors: Mostafa Ali, Naushad A. Kamar, Alireza Seif, Mohammad Maghrebi

Abstract: Open driven quantum systems have defined a powerful paradigm of non-equilibrium phases and phase transitions; however, quantum phase transitions are generically not expected in this setting due to the decohering effect of dissipation. In this work, we show that a driven-dissipative quantum spin chain exhibits a peculiar sensitivity to the ground-state quantum phase transition. Specifically, we con… ▽ More Open driven quantum systems have defined a powerful paradigm of non-equilibrium phases and phase transitions; however, quantum phase transitions are generically not expected in this setting due to the decohering effect of dissipation. In this work, we show that a driven-dissipative quantum spin chain exhibits a peculiar sensitivity to the ground-state quantum phase transition. Specifically, we consider a quantum Ising model subject to bulk dissipation (at rate $Γ$) and show that, although the correlation length remains finite (hence no phase transition), it develops a pronounced peak close to the ground-state quantum critical point. While standard techniques seem to fail in this regime, we develop a versatile analytical approach that becomes exact with vanishing dissipation ($Γ\to 0$ but finite $Γt$). On a technical level, our approach builds on previous work where the state of the system is described by a slowly evolving generalized Gibbs ensemble that accounts for the integrability of the Hamiltonian (described by free fermions) while treating dissipation perturbatively which leads to nontrivial, nonlinear equations for fermionic correlators. Finally, we demonstrate a kind of universality in that integrability-breaking perturbations of the Hamiltonian lead to the same behavior. △ Less

Submitted 30 May, 2024; originally announced May 2024.

Quantum Fourier Transform using Dynamic Circuits

Authors: Elisa Bäumer, Vinay Tripathi, Alireza Seif, Daniel Lidar, Derek S. Wang

Abstract: In dynamic quantum circuits, classical information from mid-circuit measurements is fed forward during circuit execution. This emerging capability of quantum computers confers numerous advantages that can enable more efficient and powerful protocols by drastically reducing the resource requirements for certain core algorithmic primitives. In particular, in the case of the $n$-qubit quantum Fourier… ▽ More In dynamic quantum circuits, classical information from mid-circuit measurements is fed forward during circuit execution. This emerging capability of quantum computers confers numerous advantages that can enable more efficient and powerful protocols by drastically reducing the resource requirements for certain core algorithmic primitives. In particular, in the case of the $n$-qubit quantum Fourier transform followed immediately by measurement, the scaling of resource requirements is reduced from $O(n^2)$ two-qubit gates in an all-to-all connectivity in the standard unitary formulation to $O(n)$ mid-circuit measurements in its dynamic counterpart without any connectivity constraints. Here, we demonstrate the advantage of dynamic quantum circuits for the quantum Fourier transform on IBM's superconducting quantum hardware with certified process fidelities of $>50\%$ on up to $16$ qubits and $>1\%$ on up to $37$ qubits, exceeding previous reports across all quantum computing platforms. These results are enabled by our contribution of an efficient method for certifying the process fidelity, as well as of a dynamical decoupling protocol for error suppression during mid-circuit measurements and feed-forward within a dynamic quantum circuit that we call ``feed-forward-compensated dynamical decoupling" (FC-DD). Our results demonstrate the advantages of leveraging dynamic circuits in optimizing the compilation of quantum algorithms. △ Less

Submitted 27 March, 2024; v1 submitted 14 March, 2024; originally announced March 2024.

Comments: 4 pages, 2 figures (main text) + 4 pages, 2 figures (appendix)

Suppressing Correlated Noise in Quantum Computers via Context-Aware Compiling

Authors: Alireza Seif, Haoran Liao, Vinay Tripathi, Kevin Krsulich, Moein Malekakhlagh, Mirko Amico, Petar Jurcevic, Ali Javadi-Abhari

Abstract: Coherent errors, and especially those that occur in correlation among a set of qubits, are detrimental for large-scale quantum computing. Correlations in noise can occur as a result of spatial and temporal configurations of instructions executing on the quantum processor. In this paper, we perform a detailed experimental characterization of many of these error sources, and theoretically connect th… ▽ More Coherent errors, and especially those that occur in correlation among a set of qubits, are detrimental for large-scale quantum computing. Correlations in noise can occur as a result of spatial and temporal configurations of instructions executing on the quantum processor. In this paper, we perform a detailed experimental characterization of many of these error sources, and theoretically connect them to the physics of superconducting qubits and gate operations. Equipped with this knowledge, we devise compiler strategies to suppress these errors using dynamical decoupling or error compensation into the rest of the circuit. Importantly, these strategies are successful when the context at each layer of computation is taken into account: how qubits are connected, what crosstalk terms exist on the device, and what gates or idle periods occur in that layer. Our context-aware compiler thus suppresses some dominant sources of error, making further error mitigation or error correction substantially less expensive. For example, our experiments show an increase of 18.5\% in layer fidelity for a candidate 10-qubit circuit layer compared to context-unaware suppression. Owing to the exponential nature of error mitigation, these improvements due to error suppression translate to several orders of magnitude reduction of sampling overhead for a circuit consisting of a moderate number of layers. △ Less

Submitted 26 August, 2024; v1 submitted 11 March, 2024; originally announced March 2024.

Comments: Close to published version, 16 pages, 10 figures

Journal ref: 2024 ACM/IEEE 51st Annual International Symposium on Computer Architecture (ISCA), Buenos Aires, Argentina, 2024, pp. 310-324

Demonstration of Robust and Efficient Quantum Property Learning with Shallow Shadows

Authors: Hong-Ye Hu, Andi Gu, Swarnadeep Majumder, Hang Ren, Yipei Zhang, Derek S. Wang, Yi-Zhuang You, Zlatko Minev, Susanne F. Yelin, Alireza Seif

Abstract: Extracting information efficiently from quantum systems is a major component of quantum information processing tasks. Randomized measurements, or classical shadows, enable predicting many properties of arbitrary quantum states using few measurements. While random single-qubit measurements are experimentally friendly and suitable for learning low-weight Pauli observables, they perform poorly for no… ▽ More Extracting information efficiently from quantum systems is a major component of quantum information processing tasks. Randomized measurements, or classical shadows, enable predicting many properties of arbitrary quantum states using few measurements. While random single-qubit measurements are experimentally friendly and suitable for learning low-weight Pauli observables, they perform poorly for nonlocal observables. Prepending a shallow random quantum circuit before measurements maintains this experimental friendliness, but also has favorable sample complexities for observables beyond low-weight Paulis, including high-weight Paulis and global low-rank properties such as fidelity. However, in realistic scenarios, quantum noise accumulated with each additional layer of the shallow circuit biases the results. To address these challenges, we propose the \emph{robust shallow shadows protocol}. Our protocol uses Bayesian inference to learn the experimentally relevant noise model and mitigate it in postprocessing. This mitigation introduces a bias-variance trade-off: correcting for noise-induced bias comes at the cost of a larger estimator variance. Despite this increased variance, as we demonstrate on a superconducting quantum processor, our protocol correctly recovers state properties such as expectation values, fidelity, and entanglement entropy, while maintaining a lower sample complexity compared to the random single qubit measurement scheme. We also theoretically analyze the effects of noise on sample complexity and show how the optimal choice of the shallow shadow depth varies with noise strength. This combined theoretical and experimental analysis positions the robust shallow shadow protocol as a scalable, robust, and sample-efficient protocol for characterizing quantum states on current quantum computing platforms. △ Less

Submitted 4 February, 2025; v1 submitted 27 February, 2024; originally announced February 2024.

Comments: Significant update: Added new theorems on calibration sample complexity and effective noise models. Expanded discussion on time-dependent Markovian and non-Markovian noise. Included 8 new figures presenting results on method robustness and calibration sample overhead. 28 pages and 13 figures in total

Journal ref: Nature Communications 16, 2943 (2025)

Universal Control in Bosonic Systems with Weak Kerr Nonlinearities

Authors: Ming Yuan, Alireza Seif, Andrew Lingenfelter, David I. Schuster, Aashish A. Clerk, Liang Jiang

Abstract: Resonators with weak single-photon self-Kerr nonlinearities can theoretically be used to prepare Fock states in the presence of a loss much larger than their nonlinearities. Two necessary ingredients are large displacements and a two-photon (parametric) drive. Here, we find that these systems can be controlled to achieve any desired gate operation in a finite dimensional subspace (whose dimensiona… ▽ More Resonators with weak single-photon self-Kerr nonlinearities can theoretically be used to prepare Fock states in the presence of a loss much larger than their nonlinearities. Two necessary ingredients are large displacements and a two-photon (parametric) drive. Here, we find that these systems can be controlled to achieve any desired gate operation in a finite dimensional subspace (whose dimensionality can be chosen at will). Moreover, we show that the two-photon driving requirement can be relaxed and that full controllability is achievable with only 1-photon (linear) drives. We make use of both Trotter-Suzuki decompositions and gradient-based optimization to find control pulses for a desired gate, which reduces the computational overhead by using a small blockaded subspace. We also discuss the infidelity arising from input power limitations in realistic settings, as well as from corrections to the rotating-wave approximation. Our universal control protocol opens the possibility for quantum information processing using a wide range of lossy systems with weak nonlinearities. △ Less

Submitted 25 December, 2023; originally announced December 2023.

Comments: 7 pages of main text with 3 figures. 15 pages of supplemental material with 4 figures

Journal ref: Phys. Rev. A 111, 032606 (2025)

High-fidelity, multi-qubit generalized measurements with dynamic circuits

Authors: Petr Ivashkov, Gideon Uchehara, Liang Jiang, Derek S. Wang, Alireza Seif

Abstract: Generalized measurements, also called positive operator-valued measures (POVMs), can offer advantages over projective measurements in various quantum information tasks. Here, we realize a generalized measurement of one and two superconducting qubits with high fidelity and in a single experimental setting. To do so, we propose a hybrid method, the "Naimark-terminated binary tree," based on a hybrid… ▽ More Generalized measurements, also called positive operator-valued measures (POVMs), can offer advantages over projective measurements in various quantum information tasks. Here, we realize a generalized measurement of one and two superconducting qubits with high fidelity and in a single experimental setting. To do so, we propose a hybrid method, the "Naimark-terminated binary tree," based on a hybridization of Naimark's dilation and binary tree techniques that leverages emerging hardware capabilities for mid-circuit measurements and feed-forward control. Furthermore, we showcase a highly effective use of approximate compiling to enhance POVM fidelity in noisy conditions. We argue that our hybrid method scales better toward larger system sizes than its constituent methods and demonstrate its advantage by performing detector tomography of symmetric, informationally complete POVM (SIC-POVM). Detector fidelity is further improved through a composite error mitigation strategy that incorporates twirling and a newly devised conditional readout error mitigation. Looking forward, we expect improvements in approximate compilation and hardware noise for dynamic circuits to enable generalized measurements of larger multi-qubit POVMs on superconducting qubits. △ Less

Submitted 26 August, 2024; v1 submitted 21 December, 2023; originally announced December 2023.

Comments: 15 pages, 10 figures

Journal ref: PRX Quantum 5, 030315 (2024)

Measurement and feedforward induced entanglement negativity transition

Authors: Alireza Seif, Yu-Xin Wang, Ramis Movassagh, Aashish A. Clerk

Abstract: We study the interplay between measurement-induced dynamics and conditional unitary evolution in quantum systems. We numerically and analytically investigate commuting random measurement and feedforward (MFF) processes, and find a sharp transition in their ability to generate entanglement negativity as the number of MFF channels varies. We also establish a direct connection between these findings… ▽ More We study the interplay between measurement-induced dynamics and conditional unitary evolution in quantum systems. We numerically and analytically investigate commuting random measurement and feedforward (MFF) processes, and find a sharp transition in their ability to generate entanglement negativity as the number of MFF channels varies. We also establish a direct connection between these findings and transitions induced by random dephasing from an environment with broken time-reversal symmetry. In one variant of the problem, we employ free probability theory to rigorously prove the transition's existence. Furthermore, these MFF processes have dynamic circuit representations that can be experimentally explored on current quantum computing platforms. △ Less

Submitted 25 August, 2024; v1 submitted 27 October, 2023; originally announced October 2023.

Comments: Close to final published version. 6 pages, 4 figures (main) + 14 pages, 5 figures (supplementary materials)

Journal ref: Phys. Rev. Lett. 133, 050402 (2024)

Uncovering measurement-induced entanglement via directional adaptive dynamics and incomplete information

Authors: Yu-Xin Wang, Alireza Seif, Aashish A. Clerk

Abstract: The rich entanglement dynamics and transitions exhibited by monitored quantum systems typically only exist in the conditional state, making observation extremely difficult. In this work we construct a general recipe for mimicking the conditional entanglement dynamics of a monitored system in a corresponding measurement-free dissipative system involving directional interactions between the original… ▽ More The rich entanglement dynamics and transitions exhibited by monitored quantum systems typically only exist in the conditional state, making observation extremely difficult. In this work we construct a general recipe for mimicking the conditional entanglement dynamics of a monitored system in a corresponding measurement-free dissipative system involving directional interactions between the original system and a set of auxiliary register modes. This mirror setup autonomously implements a measurement-feedforward dynamics that effectively retains a coarse-grained measurement record. We illustrate our ideas in a bosonic system featuring a competition between entangling measurements and local unitary dynamics, and also discuss extensions to qubit systems and truly many-body systems. △ Less

Submitted 20 November, 2024; v1 submitted 2 October, 2023; originally announced October 2023.

Comments: 8+13 pages, 4+9 figures

Journal ref: Phys. Rev. A 110, L050602 (2024)

Machine Learning for Practical Quantum Error Mitigation

Authors: Haoran Liao, Derek S. Wang, Iskandar Sitdikov, Ciro Salcedo, Alireza Seif, Zlatko K. Minev

Abstract: Quantum computers progress toward outperforming classical supercomputers, but quantum errors remain their primary obstacle. The key to overcoming errors on near-term devices has emerged through the field of quantum error mitigation, enabling improved accuracy at the cost of additional run time. Here, through experiments on state-of-the-art quantum computers using up to 100 qubits, we demonstrate t… ▽ More Quantum computers progress toward outperforming classical supercomputers, but quantum errors remain their primary obstacle. The key to overcoming errors on near-term devices has emerged through the field of quantum error mitigation, enabling improved accuracy at the cost of additional run time. Here, through experiments on state-of-the-art quantum computers using up to 100 qubits, we demonstrate that without sacrificing accuracy machine learning for quantum error mitigation (ML-QEM) drastically reduces the cost of mitigation. We benchmark ML-QEM using a variety of machine learning models -- linear regression, random forests, multi-layer perceptrons, and graph neural networks -- on diverse classes of quantum circuits, over increasingly complex device-noise profiles, under interpolation and extrapolation, and in both numerics and experiments. These tests employ the popular digital zero-noise extrapolation method as an added reference. Finally, we propose a path toward scalable mitigation by using ML-QEM to mimic traditional mitigation methods with superior runtime efficiency. Our results show that classical machine learning can extend the reach and practicality of quantum error mitigation by reducing its overheads and highlight its broader potential for practical quantum computations. △ Less

Submitted 22 November, 2024; v1 submitted 29 September, 2023; originally announced September 2023.

Comments: 11 pages, 7 figures (main text) + 9 pages, 4 figures (supplementary information)

Journal ref: Nature Machine Intelligence 6, 1478-1486 (2024)

Efficient multimode Wigner tomography

Authors: Kevin He, Ming Yuan, Yat Wong, Srivatsan Chakram, Alireza Seif, Liang Jiang, David I. Schuster

Abstract: Advancements in quantum system lifetimes and control have enabled the creation of increasingly complex quantum states, such as those on multiple bosonic cavity modes. When characterizing these states, traditional tomography scales exponentially in both computational and experimental measurement requirement, which becomes prohibitive as the state size increases. Here, we implement a state reconstru… ▽ More Advancements in quantum system lifetimes and control have enabled the creation of increasingly complex quantum states, such as those on multiple bosonic cavity modes. When characterizing these states, traditional tomography scales exponentially in both computational and experimental measurement requirement, which becomes prohibitive as the state size increases. Here, we implement a state reconstruction method whose sampling requirement instead scales polynomially with subspace size, and thus mode number, for states that can be expressed within such a subspace. We demonstrate this improved scaling with Wigner tomography of multimode entangled W states of up to 4 modes on a 3D circuit quantum electrodynamics (cQED) system. This approach performs similarly in efficiency to existing matrix inversion methods for 2 modes, and demonstrates a noticeable improvement for 3 and 4 modes, with even greater theoretical gains at higher mode numbers. △ Less

Submitted 18 September, 2023; originally announced September 2023.

Comments: 6 pages of main text with 4 figures. 10 pages of supplementary information with 6 figures

Realizing the Nishimori transition across the error threshold for constant-depth quantum circuits

Authors: Edward H. Chen, Guo-Yi Zhu, Ruben Verresen, Alireza Seif, Elisa Bäumer, David Layden, Nathanan Tantivasadakarn, Guanyu Zhu, Sarah Sheldon, Ashvin Vishwanath, Simon Trebst, Abhinav Kandala

Abstract: Preparing quantum states across many qubits is necessary to unlock the full potential of quantum computers. However, a key challenge is to realize efficient preparation protocols which are stable to noise and gate imperfections. Here, using a measurement-based protocol on a 127 superconducting qubit device, we study the generation of the simplest long-range order -- Ising order, familiar from Gree… ▽ More Preparing quantum states across many qubits is necessary to unlock the full potential of quantum computers. However, a key challenge is to realize efficient preparation protocols which are stable to noise and gate imperfections. Here, using a measurement-based protocol on a 127 superconducting qubit device, we study the generation of the simplest long-range order -- Ising order, familiar from Greenberger-Horne-Zeilinger (GHZ) states and the repetition code -- on 54 system qubits. Our efficient implementation of the constant-depth protocol and classical decoder shows higher fidelities for GHZ states compared to size-dependent, unitary protocols. By experimentally tuning coherent and incoherent error rates, we demonstrate stability of this decoded long-range order in two spatial dimensions, up to a critical point which corresponds to a transition belonging to the unusual Nishimori universality class. Although in classical systems Nishimori physics requires fine-tuning multiple parameters, here it arises as a direct result of the Born rule for measurement probabilities -- locking the effective temperature and disorder driving this transition. Our study exemplifies how measurement-based state preparation can be meaningfully explored on quantum processors beyond a hundred qubits. △ Less

Submitted 8 December, 2023; v1 submitted 6 September, 2023; originally announced September 2023.

Comments: 16 pages, 18 figures

Journal ref: Nature Physics 21, 161 (2025)

Efficient Long-Range Entanglement using Dynamic Circuits

Authors: Elisa Bäumer, Vinay Tripathi, Derek S. Wang, Patrick Rall, Edward H. Chen, Swarnadeep Majumder, Alireza Seif, Zlatko K. Minev

Abstract: Quantum simulation traditionally relies on unitary dynamics, inherently imposing efficiency constraints on the generation of intricate entangled states. In principle, these limitations can be superseded by non-unitary, dynamic circuits. These circuits exploit measurements alongside conditional feed-forward operations, providing a promising approach for long-range entangling gates, higher effective… ▽ More Quantum simulation traditionally relies on unitary dynamics, inherently imposing efficiency constraints on the generation of intricate entangled states. In principle, these limitations can be superseded by non-unitary, dynamic circuits. These circuits exploit measurements alongside conditional feed-forward operations, providing a promising approach for long-range entangling gates, higher effective connectivity of near-term hardware, and more efficient state preparations. Here, we explore the utility of shallow dynamic circuits for creating long-range entanglement on large-scale quantum devices. Specifically, we study two tasks: CNOT gate teleportation between up to 101 qubits by feeding forward 99 mid-circuit measurement outcomes, and the preparation of Greenberger-Horne-Zeilinger (GHZ) states with genuine entanglement. In the former, we observe that dynamic circuits can outperform their unitary counterparts. In the latter, by tallying instructions of compiled quantum circuits, we provide an error budget detailing the obstacles that must be addressed to unlock the full potential of dynamic circuits. Looking forward, we expect dynamic circuits to be useful for generating long-range entanglement in the near term on large-scale quantum devices. △ Less

Submitted 18 September, 2024; v1 submitted 24 August, 2023; originally announced August 2023.

Comments: 7 pages, 3 figures (main text) + 11 pages, 6 figures (appendix)

Journal ref: PRX Quantum 5, 030339 (2024)

Uncovering Local Integrability in Quantum Many-Body Dynamics

Authors: Oles Shtanko, Derek S. Wang, Haimeng Zhang, Nikhil Harle, Alireza Seif, Ramis Movassagh, Zlatko Minev

Abstract: Interacting many-body quantum systems and their dynamics, while fundamental to modern science and technology, are formidable to simulate and understand. However, by discovering their symmetries, conservation laws, and integrability one can unravel their intricacies. Here, using up to 124 qubits of a fully programmable quantum computer, we uncover local conservation laws and integrability in one- a… ▽ More Interacting many-body quantum systems and their dynamics, while fundamental to modern science and technology, are formidable to simulate and understand. However, by discovering their symmetries, conservation laws, and integrability one can unravel their intricacies. Here, using up to 124 qubits of a fully programmable quantum computer, we uncover local conservation laws and integrability in one- and two-dimensional periodically-driven spin lattices in a regime previously inaccessible to such detailed analysis. We focus on the paradigmatic example of disorder-induced ergodicity breaking, where we first benchmark the system crossover into a localized regime through anomalies in the one-particle-density-matrix spectrum and other hallmark signatures. We then demonstrate that this regime stems from hidden local integrals of motion by faithfully reconstructing their quantum operators, thus providing a detailed portrait of the system's integrable dynamics. Our results demonstrate a versatile strategy for extracting the hidden dynamical structure from noisy experiments on large-scale quantum computers. △ Less

Submitted 17 March, 2025; v1 submitted 14 July, 2023; originally announced July 2023.

Comments: 9 pages, 4 figures (main text) + 21 pages, 20 figures (supplementary)

Journal ref: Nat Commun 16, 2552 (2025)

Random Pulse Sequences for Qubit Noise Spectroscopy

Authors: Kaixin Huang, Demitry Farfurnik, Alireza Seif, Mohammad Hafezi, Yi-Kai Liu

Abstract: Qubit noise spectroscopy is an important tool for the experimental investigation of open quantum systems. However, conventional techniques for noise spectroscopy are time-consuming, because they require measurements of the noise spectral density at many different frequencies. Here we describe an alternative approach to noise spectroscopy, which requires fewer resources, and relies on direct measur… ▽ More Qubit noise spectroscopy is an important tool for the experimental investigation of open quantum systems. However, conventional techniques for noise spectroscopy are time-consuming, because they require measurements of the noise spectral density at many different frequencies. Here we describe an alternative approach to noise spectroscopy, which requires fewer resources, and relies on direct measurement of arbitrary linear functionals of the noise spectral density. This method uses random pulse sequences with carefully-controlled correlations, which are chosen using algorithms for phase retrieval. These measurements allow us to reconstruct sparse noise spectra via compressed sensing. Our simulations of the performance of the random pulse sequences on a realistic physical system, self-assembled quantum dots, reveal a speedup of an order of magnitude in extracting the noise spectrum, compared to conventional dynamical decoupling approaches. △ Less

Submitted 28 April, 2025; v1 submitted 1 March, 2023; originally announced March 2023.

Comments: v2: added appendix containing technical details; v3: minor clarifications; v4: minor clarifications

Journal ref: Phys. Rev. Applied 23, 054090 - Published 30 May, 2025

Simple master equations for describing driven systems subject to classical non-Markovian noise

Authors: Peter Groszkowski, Alireza Seif, Jens Koch, A. A. Clerk

Abstract: Driven quantum systems subject to non-Markovian noise are typically difficult to model even if the noise is classical. We present a systematic method based on generalized cumulant expansions for deriving a time-local master equation for such systems. This master equation has an intuitive form that directly parallels a standard Lindblad equation, but contains several surprising features: the combin… ▽ More Driven quantum systems subject to non-Markovian noise are typically difficult to model even if the noise is classical. We present a systematic method based on generalized cumulant expansions for deriving a time-local master equation for such systems. This master equation has an intuitive form that directly parallels a standard Lindblad equation, but contains several surprising features: the combination of driving and non-Markovianity results in effective time-dependent dephasing rates that can be negative, and the noise can generate Hamiltonian renormalizations even though it is classical. We analyze in detail the highly relevant case of a Rabi-driven qubit subject to various kinds of non-Markovian noise including $1/f$ fluctuations, finding an excellent agreement between our master equation and numerically-exact simulations over relevant timescales. The approach outlined here is more accurate than commonly employed phenomenological master equations which ignore the interplay between driving and noise. △ Less

Submitted 2 April, 2023; v1 submitted 8 July, 2022; originally announced July 2022.

Comments: 12+4 pages, 6+4 figures

Journal ref: Quantum 7, 972 (2023)

The learnability of Pauli noise

Authors: Senrui Chen, Yunchao Liu, Matthew Otten, Alireza Seif, Bill Fefferman, Liang Jiang

Abstract: Recently, several quantum benchmarking algorithms have been developed to characterize noisy quantum gates on today's quantum devices. A well-known issue in benchmarking is that not everything about quantum noise is learnable due to the existence of gauge freedom, leaving open the question of what information about noise is learnable and what is not, which has been unclear even for a single CNOT ga… ▽ More Recently, several quantum benchmarking algorithms have been developed to characterize noisy quantum gates on today's quantum devices. A well-known issue in benchmarking is that not everything about quantum noise is learnable due to the existence of gauge freedom, leaving open the question of what information about noise is learnable and what is not, which has been unclear even for a single CNOT gate. Here we give a precise characterization of the learnability of Pauli noise channels attached to Clifford gates, showing that learnable information corresponds to the cycle space of the pattern transfer graph of the gate set, while unlearnable information corresponds to the cut space. This implies the optimality of cycle benchmarking, in the sense that it can learn all learnable information about Pauli noise. We experimentally demonstrate noise characterization of IBM's CNOT gate up to 2 unlearnable degrees of freedom, for which we obtain bounds using physical constraints. In addition, we give an attempt to characterize the unlearnable information by assuming perfect initial state preparation. However, based on the experimental data, we conclude that this assumption is inaccurate as it yields unphysical estimates, and we obtain a lower bound on state preparation noise. △ Less

Submitted 23 December, 2022; v1 submitted 13 June, 2022; originally announced June 2022.

Comments: 33 pages, 11 figures. Close to accepted version

Journal ref: Nature Communications 14, 52 (2023)

Distributed quantum error correction for chip-level catastrophic errors

Authors: Qian Xu, Alireza Seif, Haoxiong Yan, Nam Mannucci, Bernard Ousmane Sane, Rodney Van Meter, Andrew N. Cleland, Liang Jiang

Abstract: Quantum error correction holds the key to scaling up quantum computers. Cosmic ray events severely impact the operation of a quantum computer by causing chip-level catastrophic errors, essentially erasing the information encoded in a chip. Here, we present a distributed error correction scheme to combat the devastating effect of such events by introducing an additional layer of quantum erasure err… ▽ More Quantum error correction holds the key to scaling up quantum computers. Cosmic ray events severely impact the operation of a quantum computer by causing chip-level catastrophic errors, essentially erasing the information encoded in a chip. Here, we present a distributed error correction scheme to combat the devastating effect of such events by introducing an additional layer of quantum erasure error correcting code across separate chips. We show that our scheme is fault tolerant against chip-level catastrophic errors and discuss its experimental implementation using superconducting qubits with microwave links. Our analysis shows that in state-of-the-art experiments, it is possible to suppress the rate of these errors from 1 per 10 seconds to less than 1 per month. △ Less

Submitted 30 March, 2022; originally announced March 2022.

Tailored XZZX codes for biased noise

Authors: Qian Xu, Nam Mannucci, Alireza Seif, Aleksander Kubica, Steven T. Flammia, Liang Jiang

Abstract: Quantum error correction (QEC) for generic errors is challenging due to the demanding threshold and resource requirements. Interestingly, when physical noise is biased, we can tailor our QEC schemes to the noise to improve performance. Here we study a family of codes having XZZX-type stabilizer generators, including a set of cyclic codes generalized from the five-qubit code and a set of topologica… ▽ More Quantum error correction (QEC) for generic errors is challenging due to the demanding threshold and resource requirements. Interestingly, when physical noise is biased, we can tailor our QEC schemes to the noise to improve performance. Here we study a family of codes having XZZX-type stabilizer generators, including a set of cyclic codes generalized from the five-qubit code and a set of topological codes that we call generalized toric codes (GTCs). We show that these XZZX codes are highly qubit efficient if tailored to biased noise. To characterize the code performance, we use the notion of effective distance, which generalizes code distance to the case of biased noise and constitutes a proxy for the logical failure rate. We find that the XZZX codes can achieve a favorable resource scaling by this metric under biased noise. We also show that the XZZX codes have remarkably high thresholds that reach what is achievable by random codes, and furthermore they can be efficiently decoded using matching decoders. Finally, by adding only one flag qubit, the XZZX codes can realize fault-tolerant QEC while preserving their large effective distance. In combination, our results show that tailored XZZX codes give a resource-efficient scheme for fault-tolerant QEC against biased noise. △ Less

Submitted 30 March, 2022; originally announced March 2022.

Shadow Distillation: Quantum Error Mitigation with Classical Shadows for Near-Term Quantum Processors

Authors: Alireza Seif, Ze-Pei Cian, Sisi Zhou, Senrui Chen, Liang Jiang

Abstract: Mitigating errors in quantum information processing devices is especially important in the absence of fault tolerance. An effective method in suppressing state-preparation errors is using multiple copies to distill the ideal component from a noisy quantum state. Here, we use classical shadows and randomized measurements to circumvent the need for coherent access to multiple copies at an exponentia… ▽ More Mitigating errors in quantum information processing devices is especially important in the absence of fault tolerance. An effective method in suppressing state-preparation errors is using multiple copies to distill the ideal component from a noisy quantum state. Here, we use classical shadows and randomized measurements to circumvent the need for coherent access to multiple copies at an exponential cost. We study the scaling of resources using numerical simulations and find that the overhead is still favorable compared to full state tomography. We optimize measurement resources under realistic experimental constraints and apply our method to an experiment preparing Greenberger-Horne-Zeilinger (GHZ) state with trapped ions. In addition to improving stabilizer measurements, the analysis of the improved results reveals the nature of errors affecting the experiment. Hence, our results provide a directly applicable method for mitigating errors in near-term quantum computers. △ Less

Submitted 14 March, 2022; originally announced March 2022.

Comments: 16 pages, 9 figures

Journal ref: PRX Quantum 4, 010303 (2023)

Surviving The Barren Plateau in Variational Quantum Circuits with Bayesian Learning Initialization

Authors: Ali Rad, Alireza Seif, Norbert M. Linke

Abstract: Variational quantum-classical hybrid algorithms are seen as a promising strategy for solving practical problems on quantum computers in the near term. While this approach reduces the number of qubits and operations required from the quantum machine, it places a heavy load on a classical optimizer. While often under-appreciated, the latter is a computationally hard task due to the barren plateau ph… ▽ More Variational quantum-classical hybrid algorithms are seen as a promising strategy for solving practical problems on quantum computers in the near term. While this approach reduces the number of qubits and operations required from the quantum machine, it places a heavy load on a classical optimizer. While often under-appreciated, the latter is a computationally hard task due to the barren plateau phenomenon in parameterized quantum circuits. The absence of guiding features like gradients renders conventional optimization strategies ineffective as the number of qubits increases. Here, we introduce the fast-and-slow algorithm, which uses Bayesian Learning to identify a promising region in parameter space. This is used to initialize a fast local optimizer to find the global optimum point efficiently. We illustrate the effectiveness of this method on the Bars-and-Stripes (BAS) quantum generative model, which has been studied on several quantum hardware platforms. Our results move variational quantum algorithms closer to their envisioned applications in quantum chemistry, combinatorial optimization, and quantum simulation problems. △ Less

Submitted 4 March, 2022; originally announced March 2022.

Discovering hydrodynamic equations of many-body quantum systems

Authors: Yaroslav Kharkov, Oles Shtanko, Alireza Seif, Przemyslaw Bienias, Mathias Van Regemortel, Mohammad Hafezi, Alexey V. Gorshkov

Abstract: Simulating and predicting dynamics of quantum many-body systems is extremely challenging, even for state-of-the-art computational methods, due to the spread of entanglement across the system. However, in the long-wavelength limit, quantum systems often admit a simplified description, which involves a small set of physical observables and requires only a few parameters such as sound velocity or vis… ▽ More Simulating and predicting dynamics of quantum many-body systems is extremely challenging, even for state-of-the-art computational methods, due to the spread of entanglement across the system. However, in the long-wavelength limit, quantum systems often admit a simplified description, which involves a small set of physical observables and requires only a few parameters such as sound velocity or viscosity. Unveiling the relationship between these hydrodynamic equations and the underlying microscopic theory usually requires a great effort by condensed matter theorists. In the present paper, we develop a new machine-learning framework for automated discovery of effective equations from a limited set of available data, thus bypassing complicated analytical derivations. The data can be generated from numerical simulations or come from experimental quantum simulator platforms. Using integrable models, where direct comparisons can be made, we reproduce previously known hydrodynamic equations, strikingly discover novel equations and provide their derivation whenever possible. We discover new hydrodynamic equations describing dynamics of interacting systems, for which the derivation remains an outstanding challenge. Our approach provides a new interpretable method to study properties of quantum materials and quantum simulators in non-perturbative regimes. △ Less

Submitted 3 November, 2021; originally announced November 2021.

Comments: 39 pages (11 pages main text + 28 pages supplementary material), 6+19 figures

Distinguishing between quantum and classical Markovian dephasing dissipation

Authors: Alireza Seif, Yu-Xin Wang, Aashish A. Clerk

Abstract: Understanding whether dissipation in an open quantum system is truly quantum is a question of both fundamental and practical interest. We consider n qubits subject to correlated Markovian dephasing and present a sufficient condition for when bath-induced dissipation can generate system entanglement and hence must be considered quantum. Surprisingly, we find that the presence or absence of time-rev… ▽ More Understanding whether dissipation in an open quantum system is truly quantum is a question of both fundamental and practical interest. We consider n qubits subject to correlated Markovian dephasing and present a sufficient condition for when bath-induced dissipation can generate system entanglement and hence must be considered quantum. Surprisingly, we find that the presence or absence of time-reversal symmetry plays a crucial role: broken time-reversal symmetry is required for dissipative entanglement generation. Further, simply having nonzero bath susceptibilities is not enough for the dissipation to be quantum. We also present an explicit experimental protocol for identifying truly quantum dephasing dissipation and lay the groundwork for studying more complex dissipative systems and finding optimal noise mitigating strategies. △ Less

Submitted 18 February, 2022; v1 submitted 13 September, 2021; originally announced September 2021.

Comments: 13 pages, 7 figures. Close to final published version

Journal ref: Phys. Rev. Lett. 128, 070402 (2022)

Quantum advantages for Pauli channel estimation

Authors: Senrui Chen, Sisi Zhou, Alireza Seif, Liang Jiang

Abstract: We show that entangled measurements provide an exponential advantage in sample complexity for Pauli channel estimation, which is both a fundamental problem and a practically important subroutine for benchmarking near-term quantum devices. The specific task we consider is to simultaneously learn all the eigenvalues of an $n$-qubit Pauli channel to $\pm\varepsilon$ precision. We give an estimation p… ▽ More We show that entangled measurements provide an exponential advantage in sample complexity for Pauli channel estimation, which is both a fundamental problem and a practically important subroutine for benchmarking near-term quantum devices. The specific task we consider is to simultaneously learn all the eigenvalues of an $n$-qubit Pauli channel to $\pm\varepsilon$ precision. We give an estimation protocol with an $n$-qubit ancilla that succeeds with high probability using only $O(n/\varepsilon^{2})$ copies of the Pauli channel, while prove that any ancilla-free protocol (possibly with adaptive control and channel concatenation) would need at least $Ω(2^{n/3})$ rounds of measurement. We further study the advantages provided by a small number of ancillas. For the case that a $k$-qubit ancilla ($k\le n$) is available, we obtain a sample complexity lower bound of $Ω(2^{(n-k)/3})$ for any non-concatenating protocol, and a stronger lower bound of $Ω(n2^{n-k})$ for any non-adaptive, non-concatenating protocol, which is shown to be tight. We also show how to apply the ancilla-assisted estimation protocol to a practical quantum benchmarking task in a noise-resilient and sample-efficient manner, given reasonable noise assumptions. Our results provide a practically-interesting example for quantum advantages in learning and also bring new insight for quantum benchmarking. △ Less

Submitted 8 November, 2021; v1 submitted 19 August, 2021; originally announced August 2021.

Comments: 21 pages, 5 figures. Introduction rewritten, additional references added, typo corrected

Journal ref: Phys. Rev. A 105, 032435 (2022)

Engineering an Effective Three-spin Hamiltonian in Trapped-ion Systems for Applications in Quantum Simulation

Authors: Bárbara Andrade, Zohreh Davoudi, Tobias Graß, Mohammad Hafezi, Guido Pagano, Alireza Seif

Abstract: Trapped-ion quantum simulators, in analog and digital modes, are considered a primary candidate to achieve quantum advantage in quantum simulation and quantum computation. The underlying controlled ion-laser interactions induce all-to-all two-spin interactions via the collective modes of motion through Cirac-Zoller or Molmer-Sorensen schemes, leading to effective two-spin Hamiltonians, as well as… ▽ More Trapped-ion quantum simulators, in analog and digital modes, are considered a primary candidate to achieve quantum advantage in quantum simulation and quantum computation. The underlying controlled ion-laser interactions induce all-to-all two-spin interactions via the collective modes of motion through Cirac-Zoller or Molmer-Sorensen schemes, leading to effective two-spin Hamiltonians, as well as two-qubit entangling gates. In this work, the Molmer-Sorensen scheme is extended to induce three-spin interactions via tailored first- and second-order spin-motion couplings. The scheme enables engineering single-, two-, and three-spin interactions, and can be tuned via an enhanced protocol to simulate purely three-spin dynamics. Analytical results for the effective evolution are presented, along with detailed numerical simulations of the full dynamics to support the accuracy and feasibility of the proposed scheme for near-term applications. With a focus on quantum simulation, the advantage of a direct analog implementation of three-spin dynamics is demonstrated via the example of matter-gauge interactions in the U(1) lattice gauge theory within the quantum link model. The mapping of degrees of freedom and strategies for scaling the three-spin scheme to larger systems, are detailed, along with a discussion of the expected outcome of the simulation of the quantum link model given realistic fidelities in the upcoming experiments. The applications of the three-spin scheme go beyond the lattice gauge theory example studied here and include studies of static and dynamical phase diagrams of strongly interacting condensed-matter systems modeled by two- and three-spin Hamiltonians. △ Less

Submitted 2 August, 2021; originally announced August 2021.

Comments: 24 pages, 9 figures, 3 tables

Report number: UMD-PP-021-05

Journal ref: Quantum Sci. Technol. 7 034001 (2022)

Decoding conformal field theories: from supervised to unsupervised learning

Authors: En-Jui Kuo, Alireza Seif, Rex Lundgren, Seth Whitsitt, Mohammad Hafezi

Abstract: We use machine learning to classify rational two-dimensional conformal field theories. We first use the energy spectra of these minimal models to train a supervised learning algorithm. We find that the machine is able to correctly predict the nature and the value of critical points of several strongly correlated spin models using only their energy spectra. This is in contrast to previous works tha… ▽ More We use machine learning to classify rational two-dimensional conformal field theories. We first use the energy spectra of these minimal models to train a supervised learning algorithm. We find that the machine is able to correctly predict the nature and the value of critical points of several strongly correlated spin models using only their energy spectra. This is in contrast to previous works that use machine learning to classify different phases of matter, but do not reveal the nature of the critical point between phases. Given that the ground-state entanglement Hamiltonian of certain topological phases of matter is also described by conformal field theories, we use supervised learning on Réyni entropies and find that the machine is able to identify which conformal field theory describes the entanglement Hamiltonian with only the lowest few Réyni entropies to a high degree of accuracy. Finally, using autoencoders, an unsupervised learning algorithm, we find a hidden variable that has a direct correlation with the central charge and discuss prospects for using machine learning to investigate other conformal field theories, including higher-dimensional ones. Our results highlight that machine learning can be used to find and characterize critical points and also hint at the intriguing possibility to use machine learning to learn about more complex conformal field theories. △ Less

Submitted 10 July, 2021; v1 submitted 25 June, 2021; originally announced June 2021.

Compressed Sensing Measurement of Long-Range Correlated Noise

Authors: Alireza Seif, Mohammad Hafezi, Yi-Kai Liu

Abstract: Long-range correlated errors can severely impact the performance of NISQ (noisy intermediate-scale quantum) devices, and fault-tolerant quantum computation. Characterizing these errors is important for improving the performance of these devices, via calibration and error correction, and to ensure correct interpretation of the results. We propose a compressed sensing method for detecting two-qubit… ▽ More Long-range correlated errors can severely impact the performance of NISQ (noisy intermediate-scale quantum) devices, and fault-tolerant quantum computation. Characterizing these errors is important for improving the performance of these devices, via calibration and error correction, and to ensure correct interpretation of the results. We propose a compressed sensing method for detecting two-qubit correlated dephasing errors, assuming only that the correlations are sparse (i.e., at most s pairs of qubits have correlated errors, where s << n(n-1)/2, and n is the total number of qubits). In particular, our method can detect long-range correlations between any two qubits in the system (i.e., the correlations are not restricted to be geometrically local). Our method is highly scalable: it requires as few as m = O(s log n) measurement settings, and efficient classical postprocessing based on convex optimization. In addition, when m = O(s log^4(n)), our method is highly robust to noise, and has sample complexity O(max(n,s)^2 log^4(n)), which can be compared to conventional methods that have sample complexity O(n^3). Thus, our method is advantageous when the correlations are sufficiently sparse, that is, when s < O(n^(3/2) / log^2(n)). Our method also performs well in numerical simulations on small system sizes, and has some resistance to state-preparation-and-measurement (SPAM) errors. The key ingredient in our method is a new type of compressed sensing measurement, which works by preparing entangled Greenberger-Horne-Zeilinger states (GHZ states) on random subsets of qubits, and measuring their decay rates with high precision. △ Less

Submitted 26 May, 2021; originally announced May 2021.

Comments: 38 pages, 6 figures

Meta Hamiltonian Learning

Authors: Przemyslaw Bienias, Alireza Seif, Mohammad Hafezi

Abstract: Efficient characterization of quantum devices is a significant challenge critical for the development of large scale quantum computers. We consider an experimentally motivated situation, in which we have a decent estimate of the Hamiltonian, and its parameters need to be characterized and fine-tuned frequently to combat drifting experimental variables. We use a machine learning technique known as… ▽ More Efficient characterization of quantum devices is a significant challenge critical for the development of large scale quantum computers. We consider an experimentally motivated situation, in which we have a decent estimate of the Hamiltonian, and its parameters need to be characterized and fine-tuned frequently to combat drifting experimental variables. We use a machine learning technique known as meta-learning to learn a more efficient optimizer for this task. We consider training with the nearest-neighbor Ising model and study the trained model's generalizability to other Hamiltonian models and larger system sizes. We observe that the meta-optimizer outperforms other optimization methods in average loss over test samples. This advantage follows from the meta-optimizer being less likely to get stuck in local minima, which highly skews the distribution of the final loss of the other optimizers. In general, meta-learning decreases the number of calls to the experiment and reduces the needed classical computational resources. △ Less

Submitted 9 April, 2021; originally announced April 2021.

Entanglement entropy scaling transition under competing monitoring protocols

Authors: Mathias Van Regemortel, Ze-Pei Cian, Alireza Seif, Hossein Dehghani, Mohammad Hafezi

Abstract: Dissipation generally leads to the decoherence of a quantum state. In contrast, numerous recent proposals have illustrated that dissipation can also be tailored to stabilize many-body entangled quantum states. While the focus of these works has been primarily on engineering the non-equilibrium steady state, we investigate the build-up of entanglement in the quantum trajectories. Specifically, we a… ▽ More Dissipation generally leads to the decoherence of a quantum state. In contrast, numerous recent proposals have illustrated that dissipation can also be tailored to stabilize many-body entangled quantum states. While the focus of these works has been primarily on engineering the non-equilibrium steady state, we investigate the build-up of entanglement in the quantum trajectories. Specifically, we analyze the competition between two different dissipation channels arising from two incompatible continuous monitoring protocols. The first protocol locks the phase of neighboring sites upon registering a quantum jump, thereby generating a long-range entanglement through the system, while the second destroys the coherence via a dephasing mechanism. By studying the unraveling of stochastic quantum trajectories associated with the continuous monitoring protocols, we present a transition for the scaling of the averaged trajectory entanglement entropies, from critical scaling to area-law behavior. Our work provides an alternative perspective on the measurement-induced phase transition: the measurement can be viewed as monitoring and registering quantum jumps, offering an intriguing extension of these phase transitions through the long-established realm of quantum optics. △ Less

Submitted 25 February, 2021; v1 submitted 19 August, 2020; originally announced August 2020.

Journal ref: Phys. Rev. Lett. 126, 123604 (2021)

Optimal control for quantum detectors

Authors: Paraj Titum, Kevin M. Schultz, Alireza Seif, Gregory D. Quiroz, B. D. Clader

Abstract: Quantum systems are promising candidates for sensing of weak signals as they can provide unrivaled performance when estimating parameters of external fields. However, when trying to detect weak signals that are hidden by background noise, the signal-to-noise-ratio is a more relevant metric than raw sensitivity. We identify, under modest assumptions about the statistical properties of the signal an… ▽ More Quantum systems are promising candidates for sensing of weak signals as they can provide unrivaled performance when estimating parameters of external fields. However, when trying to detect weak signals that are hidden by background noise, the signal-to-noise-ratio is a more relevant metric than raw sensitivity. We identify, under modest assumptions about the statistical properties of the signal and noise, the optimal quantum control to detect an external signal in the presence of background noise using a quantum sensor. Interestingly, for white background noise, the optimal solution is the simple and well-known spin-locking control scheme. We further generalize, using numerical techniques, these results to the background noise being a correlated Lorentzian spectrum. We show that for increasing correlation time, pulse based sequences such as CPMG are also close to the optimal control for detecting the signal, with the crossover dependent on the signal frequency. These results show that an optimal detection scheme can be easily implemented in near-term quantum sensors without the need for complicated pulse shaping. △ Less

Submitted 12 May, 2020; originally announced May 2020.

Journal ref: npj Quantum Inf 7, 53 (2021)

Machine learning the thermodynamic arrow of time

Authors: Alireza Seif, Mohammad Hafezi, Christopher Jarzynski

Abstract: The mechanism by which thermodynamics sets the direction of time's arrow has long fascinated scientists. Here, we show that a machine learning algorithm can learn to discern the direction of time's arrow when provided with a system's microscopic trajectory as input. The performance of our algorithm matches fundamental bounds predicted by nonequilibrium statistical mechanics. Examination of the alg… ▽ More The mechanism by which thermodynamics sets the direction of time's arrow has long fascinated scientists. Here, we show that a machine learning algorithm can learn to discern the direction of time's arrow when provided with a system's microscopic trajectory as input. The performance of our algorithm matches fundamental bounds predicted by nonequilibrium statistical mechanics. Examination of the algorithm's decision-making process reveals that it discovers the underlying thermodynamic mechanism and the relevant physical observables. Our results indicate that machine learning techniques can be used to study systems out of equilibrium, and ultimately to uncover physical principles. △ Less

Submitted 26 September, 2019; originally announced September 2019.

Comments: 15 pages, 10 figures

Towards analog quantum simulations of lattice gauge theories with trapped ions

Authors: Zohreh Davoudi, Mohammad Hafezi, Christopher Monroe, Guido Pagano, Alireza Seif, Andrew Shaw

Abstract: Gauge field theories play a central role in modern physics and are at the heart of the Standard Model of elementary particles and interactions. Despite significant progress in applying classical computational techniques to simulate gauge theories, it has remained a challenging task to compute the real-time dynamics of systems described by gauge theories. An exciting possibility that has been explo… ▽ More Gauge field theories play a central role in modern physics and are at the heart of the Standard Model of elementary particles and interactions. Despite significant progress in applying classical computational techniques to simulate gauge theories, it has remained a challenging task to compute the real-time dynamics of systems described by gauge theories. An exciting possibility that has been explored in recent years is the use of highly-controlled quantum systems to simulate, in an analog fashion, properties of a target system whose dynamics are difficult to compute. Engineered atom-laser interactions in a linear crystal of trapped ions offer a wide range of possibilities for quantum simulations of complex physical systems. Here, we devise practical proposals for analog simulation of simple lattice gauge theories whose dynamics can be mapped onto spin-spin interactions in any dimension. These include 1+1D quantum electrodynamics, 2+1D Abelian Chern-Simons theory coupled to fermions, and 2+1D pure Z2 gauge theory. The scheme proposed, along with the optimization protocol applied, will have applications beyond the examples presented in this work, and will enable scalable analog quantum simulation of Heisenberg spin models in any number of dimensions and with arbitrary interaction strengths. △ Less

Submitted 8 August, 2019; originally announced August 2019.

Comments: 27 pages, 15 figures. 1 Mathematica Notebook as ancillary file

Report number: UMD-PP-019-03

Journal ref: Phys. Rev. Research 2, 023015 (2020)

Photon pair condensation by engineered dissipation

Authors: Ze-Pei Cian, Guanyu Zhu, Su-Kuan Chu, Alireza Seif, Wade DeGottardi, Liang Jiang, Mohammad Hafezi

Abstract: Dissipation can usually induce detrimental decoherence in a quantum system. However, engineered dissipation can be used to prepare and stabilize coherent quantum many-body states. Here, we show that by engineering dissipators containing photon pair operators, one can stabilize an exotic dark state, which is a condensate of photon pairs with a phase-nematic order. In this system, the usual superflu… ▽ More Dissipation can usually induce detrimental decoherence in a quantum system. However, engineered dissipation can be used to prepare and stabilize coherent quantum many-body states. Here, we show that by engineering dissipators containing photon pair operators, one can stabilize an exotic dark state, which is a condensate of photon pairs with a phase-nematic order. In this system, the usual superfluid order parameter, i.e. single-photon correlation, is absent, while the photon pair correlation exhibits long-range order. Although the dark state is not unique due to multiple parity sectors, we devise an additional type of dissipators to stabilize the dark state in a particular parity sector via a diffusive annihilation process which obeys Glauber dynamics in an Ising model. Furthermore, we propose an implementation of these photon-pair dissipators in circuit-QED architecture. △ Less

Submitted 6 July, 2019; v1 submitted 29 March, 2019; originally announced April 2019.

Journal ref: Phys. Rev. Lett. 123, 063602 (2019)

Machine learning assisted readout of trapped-ion qubits

Authors: Alireza Seif, Kevin A. Landsman, Norbert M. Linke, Caroline Figgatt, C. Monroe, Mohammad Hafezi

Abstract: We reduce measurement errors in a quantum computer using machine learning techniques. We exploit a simple yet versatile neural network to classify multi-qubit quantum states, which is trained using experimental data. This flexible approach allows the incorporation of any number of features of the data with minimal modifications to the underlying network architecture. We experimentally illustrate t… ▽ More We reduce measurement errors in a quantum computer using machine learning techniques. We exploit a simple yet versatile neural network to classify multi-qubit quantum states, which is trained using experimental data. This flexible approach allows the incorporation of any number of features of the data with minimal modifications to the underlying network architecture. We experimentally illustrate this approach in the readout of trapped-ion qubits using additional spatial and temporal features in the data. Using this neural network classifier, we efficiently treat qubit readout crosstalk, resulting in a 30\% improvement in detection error over the conventional threshold method. Our approach does not depend on the specific details of the system and can be readily generalized to other quantum computing platforms. △ Less

Submitted 1 May, 2018; v1 submitted 20 April, 2018; originally announced April 2018.

Journal ref: J. Phys. B: At. Mol. Opt. Phys. 51 174006 (2018)

Thermal management and non-reciprocal control of phonon flow via optomechanics

Authors: Alireza Seif, Wade DeGottardi, Keivan Esfarjani, Mohammad Hafezi

Abstract: Engineering phonon transport in physical systems is a subject of interest in the study of materials and plays a crucial role in controlling energy and heat transfer. Of particular interest are non-reciprocal phononic systems, which in direct analogy to electric diodes, provide a directional flow of energy. Here, we propose an engineered nanostructured material, in which tunable non-reciprocal phon… ▽ More Engineering phonon transport in physical systems is a subject of interest in the study of materials and plays a crucial role in controlling energy and heat transfer. Of particular interest are non-reciprocal phononic systems, which in direct analogy to electric diodes, provide a directional flow of energy. Here, we propose an engineered nanostructured material, in which tunable non-reciprocal phonon transport is achieved through optomechanical coupling. Our scheme relies on breaking time-reversal symmetry by a spatially varying laser drive, which manipulates low-energy acoustic phonons. Furthermore, we take advantage of recent developments in the manipulation of high-energy phonons through controlled scattering mechanisms, such as using alloys and introducing disorder. These combined approaches allow us to design an acoustic isolator and a thermal diode. Our proposed device will have potential impact in phonon-based information processing, and heat management in low temperatures. △ Less

Submitted 23 March, 2018; v1 submitted 24 October, 2017; originally announced October 2017.

Comments: 7 pages, 3 figure, and supplementary material

Journal ref: Nat. Commun. 9, 1207 (2018)

Measurement Protocol for the Entanglement Spectrum of Cold Atoms

Authors: Hannes Pichler, Guanyu Zhu, Alireza Seif, Peter Zoller, Mohammad Hafezi

Abstract: Entanglement, and, in particular the entanglement spectrum, plays a major role in characterizing many-body quantum systems. While there has been a surge of theoretical works on the subject, no experimental measurement has been performed to date because of the lack of an implementable measurement scheme. Here, we propose a measurement protocol to access the entanglement spectrum of many-body states… ▽ More Entanglement, and, in particular the entanglement spectrum, plays a major role in characterizing many-body quantum systems. While there has been a surge of theoretical works on the subject, no experimental measurement has been performed to date because of the lack of an implementable measurement scheme. Here, we propose a measurement protocol to access the entanglement spectrum of many-body states in experiments with cold atoms in optical lattices. Our scheme effectively performs a Ramsey spectroscopy of the entanglement Hamiltonian and is based on the ability to produce several copies of the state under investigation together with the possibility to perform a global swap gate between two copies conditioned on the state of an auxiliary qubit. We show how the required conditional swap gate can be implemented with cold atoms, either by using Rydberg interactions or coupling the atoms to a cavity mode. We illustrate these ideas on a simple (extended) Bose-Hubbard model where such a measurement protocol reveals topological features of the Haldane phase. △ Less

Submitted 22 November, 2016; v1 submitted 27 May, 2016; originally announced May 2016.

Journal ref: Phys. Rev. X 6, 041033 (2016)