Quantum Overlapping Tomography

Cotler, Jordan; Wilczek, Frank

doi:10.1103/physrevlett.124.100401

Cited by 124 publications

(119 citation statements)

References 59 publications

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“…Additionally, we demonstrate that one can leverage the anticommuting structure of fermionic systems by constructing such sets of size 1 ≤ ω ≤ N to measure all 4-Majorana operators in OðN 4 =ωÞ time with a gate count and circuit depth of only ω, allowing one to trade off a decrease in coherence time requirements for an increase in the number of measurements required. We note that during the final stages of preparing this manuscript, a preprint was posted to arXiv which independently develops a similar scheme for measuring k-qubit RDMs [19]. This scheme seems to be identical to ours for k ¼ 2 but uses insights about hash functions to generalize the scheme to higher k with scaling of e OðkÞ log N, which improves over our bound of Oð3 k log k−1 NÞ by polylogarithmic factors in N.…”

Section: Discussionmentioning

confidence: 61%

“…For instance, the fermionic 2-RDM allows one to calculate such properties as energy [14], energy gradients [7,15], and multipole moments [16] of electronic systems in quantum chemistry and condensed matter problems, and further enables techniques for relaxing orbitals to reduce basis error [5,17]. By contrast, the qubit 2-RDM plays a vital role in spin systems, as it contains static spin correlation functions that can be used to predict phases and phase transitions [18], and separately contains information to characterize the entanglement generated on a quantum device [19]. Reduced density matrices thus offer a useful and tractable description of an otherwise complex quantum state.…”

Section: Introductionmentioning

confidence: 99%

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Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States

2020

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Many applications of quantum simulation require one to prepare and then characterize quantum states by efficiently estimating k-body reduced density matrices (k-RDMs), from which observables of interest may be obtained. For instance, the fermionic 2-RDM contains the energy, charge density, and energy gradients of an electronic system, while the qubit 2-RDM contains the spatial correlation functions of magnetic systems. Naive estimation of such RDMs requires repeated state preparations for each matrix element, which makes for prohibitively large computation times. However, commuting matrix elements may be measured simultaneously, allowing for a significant cost reduction. In this work, we design schemes for such a parallelization with near-optimal complexity in the system size N. We first describe a scheme to sample all elements of a qubit k-RDM using only Oð3 k log k−1 NÞ unique measurement circuits, an exponential improvement over prior art. We then describe a scheme for sampling all elements of the fermionic 2-RDM using only OðN 2 Þ unique measurement circuits, each of which requires only a local OðNÞ-depth measurement circuit. We prove a lower bound of Ωðϵ −2 N k Þ on the number of state preparations, Clifford circuits, and measurement in the computational basis required to estimate all elements of a fermionic k-RDM, making our scheme for sampling the fermionic 2-RDM asymptotically optimal. We finally construct circuits to sample the expectation value of a linear combination of ω anticommuting two-body fermionic operators with only OðωÞ gates on a linear array. These circuits allows for sampling any linear combination of fermionic 2-RDM elements in OðN 4 =ωÞ time, with a significantly lower measurement circuit complexity than prior art. Our results improve the viability of near-term quantum simulation of molecules and strongly correlated material systems.

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Section: Discussionmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States

2020

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“…We show how to efficiently characterize the noise using a number of circuits scaling logarithmically with the number of qubits. To this aim, we generalize the techniques of the recently introduced Quantum Overlapping Tomography (QOT) [33] to the problem of readout noise reconstruction. Specifically, we introduce notion of Diagonal Detector Overlapping Tomography (DDOT) which allows to reconstruct noise description with k-local cross-talk on N -qubit device using O k2 k log (N ) quantum circuits consisting of single layer of X and identity gates.…”

Section: Results Outlinementioning

confidence: 99%

“…However, for any characterization procedure, it is expedient to utilize as few resources as possible. In order to reduce the number of circuits even further, in the next Section we generalize the recently introduced Quantum Overlapping Tomography (QOT) [33] (see also recent followups [46,47]) to the context of Diagonal Detector Tomography. We will refer to our method as Diagonal Detector Overlapping Tomography (DDOT).…”

Section: Characterization Of Readout Noisementioning

confidence: 99%

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Modeling and mitigation of cross-talk effects in readout noise with applications to the Quantum Approximate Optimization Algorithm

Maciejewski

Baccari

Zimborás

et al. 2021

Quantum

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Measurement noise is one of the main sources of errors in currently available quantum devices based on superconducting qubits. At the same time, the complexity of its characterization and mitigation often exhibits exponential scaling with the system size. In this work, we introduce a correlated measurement noise model that can be efficiently described and characterized, and which admits effective noise-mitigation on the level of marginal probability distributions. Noise mitigation can be performed up to some error for which we derive upper bounds. Characterization of the model is done efficiently using Diagonal Detector Overlapping Tomography – a generalization of the recently introduced Quantum Overlapping Tomography to the problem of reconstruction of readout noise with restricted locality. The procedure allows to characterize k-local measurement cross-talk on N-qubit device using O(k2klog(N)) circuits containing random combinations of X and identity gates. We perform experiments on 15 (23) qubits using IBM's (Rigetti's) devices to test both the noise model and the error-mitigation scheme, and obtain an average reduction of errors by a factor >22 (>5.5) compared to no mitigation. Interestingly, we find that correlations in the measurement noise do not correspond to the physical layout of the device. Furthermore, we study numerically the effects of readout noise on the performance of the Quantum Approximate Optimization Algorithm (QAOA). We observe in simulations that for numerous objective Hamiltonians, including random MAX-2-SAT instances and the Sherrington-Kirkpatrick model, the noise-mitigation improves the quality of the optimization. Finally, we provide arguments why in the course of QAOA optimization the estimates of the local energy (or cost) terms often behave like uncorrelated variables, which greatly reduces sampling complexity of the energy estimation compared to the pessimistic error analysis. We also show that similar effects are expected for Haar-random quantum states and states generated by shallow-depth random circuits.

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Measurements of Quantum Hamiltonians with Locally-Biased Classical Shadows

Hadfield

Bravyi

Raymond

et al. 2022

Commun. Math. Phys.

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Obtaining precise estimates of quantum observables is a crucial step of variational quantum algorithms. We consider the problem of estimating expectation values of quantum Hamiltonians, obtained on states prepared on a quantum computer. We propose a novel estimator for this task, which is locally optimised with knowledge of the Hamiltonian and a classical approximation to the underlying quantum state. Our estimator is based on the concept of classical shadows of a quantum state, and has the important property of not adding to the circuit depth for the state preparation. We test its performance numerically for molecular Hamiltonians of increasing size, finding a sizable reduction in variance with respect to current measurement protocols that do not increase circuit depths.

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Quantum Overlapping Tomography

Cited by 124 publications

References 59 publications

Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States

Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States

Modeling and mitigation of cross-talk effects in readout noise with applications to the Quantum Approximate Optimization Algorithm

Measurements of Quantum Hamiltonians with Locally-Biased Classical Shadows

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