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arXiv:2301.07109

Benchmarking multi-qubit gates -- II: Computational aspects

Bharath Hebbe Madhusudhana

Published Jan 17, 2023• 0 citations
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Abstract

An important step in developing multi-qubit gates is to construct efficient benchmarking protocols for them. In our previous paper (arXiv: 2210.04330), we developed metrological protocols to measure the reduced Choi matrix i.e., the completely positive (CP) maps induced on a subset S of the qubits, by the multi-qubit gate. Here, we show a set of classically verifiable properties that the Choi matrix satisfies if it is a reduction of a multi-qubit unitary and use them to develop benchmarks. We identify three types of errors that affect the implementation of a multi-qubit unitary, based on their mathematical properties and physical origin. Although a target multi-qubit gate is a unitary operator, errors turn it into a general completely positive (CP) map. Errors due to coupling to a thermal bath result in the multi-qubit gate being CP-divisible (Markovian), deviating from a unitary. The reduced Choi matrix of a multi-qubit gate has a property known as double stochasticity, which is violated in the presence of Markovian errors. We construct a benchmark using double-stochasticity violation and show that it is sensitive to coupling to any thermal bath at a finite temperature. Further, errors due to shot-to-shot fluctuations result in a non-markovian, i.e., CP-indivisible quantum process. We prove a new property, which we call the \rank property of the reduced Choi matrix, the violation of which implies a CP-indivisible error. A third category of errors comes from systematics in the implementation of a multi-qubit gate, resulting in no deviation from unitarity. We refer to this as unitary errors. This corresponds to the most challenging type of error to benchmark. We develop a partial-benchmarking protocol for such errors using symmetries of the multi-qubit gate being applied.

DOI
10.48550/arxiv.2301.07109

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An important step in developing multi-qubit gates is to construct efficient benchmarking protocols for them. In our previous paper (arXiv: 2210.04330), we developed metrological protocols to measure the reduced Choi matrix i.e., the completely positive (CP) maps induced on a subset S of the qubits, by the multi-qubit gate. Here, we show a set of classically verifiable properties that the Choi matrix satisfies if it is a reduction of a multi-qubit unitary and use them to develop benchmarks. We identify three types of errors that affect the implementation of a multi-qubit unitary, based on their mathematical properties and physical origin. Although a target multi-qubit gate is a unitary operator, errors turn it into a general completely positive (CP) map. Errors due to coupling to a thermal bath result in the multi-qubit gate being CP-divisible (Markovian), deviating from a unitary. The reduced Choi matrix of a multi-qubit gate has a property known as double stochasticity, which is violated in the presence of Markovian errors. We construct a benchmark using double-stochasticity violation and show that it is sensitive to coupling to any thermal bath at a finite temperature. Further, errors due to shot-to-shot fluctuations result in a non-markovian, i.e., CP-indivisible quantum process. We prove a new property, which we call the \rank property of the reduced Choi matrix, the violation of which implies a CP-indivisible error. A third category of errors comes from systematics in the implementation of a multi-qubit gate, resulting in no deviation from unitarity. We refer to this as unitary errors. This corresponds to the most challenging type of error to benchmark. We develop a partial-benchmarking protocol for such errors using symmetries of the multi-qubit gate being applied.
Accurate and precise control of large quantum systems is paramount to achieve practical advantages on quantum devices. Therefore, benchmarking the hardware errors in quantum computers has drawn significant attention lately. Existing benchmarks for digital quantum computers involve averaging the global fidelity over a large set of quantum circuits and are therefore unsuitable for specific multi-qubit gates used in analog quantum operations. Moreover, average global fidelity is not the optimal figure-of-merit for some of the applications specific to multi-qubit gates and analog devices , such as the study of many-body physics, which often use local observables. In this two-part paper, we develop a new figure-of-merit suitable for multi-qubit quantum gates based on the reduced Choi matrix of the operation. In the first part, we develop an efficient, scalable protocol to completely characterize the reduced Choi matrix. We identify two sources of sampling errors in measurements of the reduced Choi matrix and we show that there are fundamental limits to the rate of convergence of the sampling errors, analogous to the standard quantum limit and Heisenberg limit. A slow convergence rate of sampling errors would mean that we need a large number of experimental shots. We develop protocols using quantum information scrambling, which has been observed in disordered systems for e.g., to speed up the rate of convergence of the sampling error at state preparation Moreover, we develop protocols using squeezed and entangled initial states to enhance the convergence rate of the sampling error at measurement, which results in a metrologically enhanced reduced process tomography protocol.
Quantum simulation of many-body systems, particularly using ultracold atoms and trapped ions, presents a unique form of quantum control—it is a direct implementation of a multi-qubit gate generated by the Hamiltonian. As a consequence, it also faces a unique challenge in terms of benchmarking, because the well-established gate benchmarking techniques are unsuitable for this form of quantum control. Here we show that the symmetries of the target many-body Hamiltonian can be used not only to benchmark but to characterize experimental errors in the quantum simulation. We use our results to develop protocols to characterize these errors, which can be implemented using state-of-the-art technology. We consider two forms of errors: (i) unitary errors arising out of systematic errors in the applied Hamiltonian and (ii) canonical non-Markovian errors arising out of random shot-to-shot fluctuations in the applied Hamiltonian. We show that the dynamics of the expectation value of the target Hamiltonian itself, which is ideally constant in time, can be used to characterize these errors. In the presence of errors, the expectation value of the target Hamiltonian shows a characteristic thermalization dynamics, when it satisfies the operator thermalization hypothesis (OTH). That is, an oscillation in the short time followed by relaxation to a steady-state value in the long time limit. We show that while the steady-state value can be used to characterize the coherent errors, the amplitude of the oscillations can be used to estimate the non-Markovian errors. We prove a sandwich theorem to establish a linear relation between the amplitude of the oscillations and the magnitude of the non-Markovian errors. Moreover, by varying the initial state, we show that the steady state values can be used to completely construct the generator of the coherent errors. Using these results, we develop two experimental protocols to characterize the unitary errors based on these results, one of which requires single-qubit addressing and the other one doesn't. We also develop a protocol to characterize non-Markovian errors. Published by the American Physical Society 2024
In non-Markovian systems, the current state of the system depends on the full or partial history of its past evolution. Owing to these time correlations, non-Markovian noise violates common assumptions in gate characterization protocols such as randomized benchmarking and gate-set tomography. Here, we perform a case study of the effects of a quantum non-Markovian bath on qubit randomized benchmarking experiments. We consider a model consisting of qubits coupled to a multimode bosonic bath. We apply unitary operations on the qubits, interspersed with brief interactions with the environment governed by a Hamiltonian. Allowing for non-Markovianity in the interactions leads to clear differences in the randomized benchmarking decay curves in this model, which we analyze in detail. The Markovian model’s decay is exponential as expected, whereas the model with non-Markovian interactions displays a much slower, almost polynomial, decay. We develop efficient numerical methods for this problem that we use to support our theoretical results. These results inform efforts on incorporating quantum non-Markovian noise in the characterization and benchmarking of quantum devices.
Characterization of experimental systems is an essential step in developing and improving quantum hardware. A collection of protocols known as Randomized Benchmarking (RB) was developed in the past decade, which provides an efficient way to measure error rates in quantum systems. In a recent paper (arxiv:2010.07974), a general framework for RB was proposed, which encompassed most of the known RB protocols and overcame the limitation on error models in previous works. However, even this general framework has a restriction: it can only be applied to a finite group of gates. This does not meet the need posed by experiments, in particular the demand for benchmarking non-Clifford gates and continuous gate sets on quantum devices. In this work we generalize the RB framework to continuous groups of gates and show that as long as the noise level is reasonably small, the output can be approximated as a linear combination of matrix exponential decays. As an application, we numerically study the fully randomized benchmarking protocol (i.e. RB with the entire unitary group as the gate set) enabled by our proof. This provides a unified way to estimate the gate fidelity for any quantum gate in an experiment.
With the development of controllable quantum systems, fast and practical characterization of multi-qubit gates has become essential for building high-fidelity quantum computing devices. The usual way to fulfill this requirement via randomized benchmarking demands complicated implementation of numerous multi-qubit twirling gates. How to efficiently and reliably estimate the fidelity of a quantum process remains an open problem. This work thus proposes a character-cycle benchmarking protocol and a character-average benchmarking protocol using only local twirling gates to estimate the process fidelity of an individual multi-qubit operation. Our protocols were able to characterize a large class of quantum gates including and beyond the Clifford group via the local gauge transformation, which forms a universal gate set for quantum computing. We demonstrated numerically our protocols for a non-Clifford gate—controlled- ( T X ) and a Clifford gate—five-qubit quantum error-correcting encoding circuit. The numerical results show that our protocols can efficiently and reliably characterize the gate process fidelities. Compared with the cross-entropy benchmarking, the simulation results show that the character-average benchmarking achieves three orders of magnitude improvements in terms of sampling complexity.
We study a system qubit repeatedly interacting with the same environmental qubit, with a reservoir acting on the environment between collisions via a completely positive, trace-preserving map. We show that complete suppression of system--environment correlations uniquely requires a full environmental reset, recovering a semi group dynamics with a time-independent Gorini--Kossakowski--Sudarshan--Lindblad generator, whereas a partial reset yields a continuous transition between Markovian and non-Markovian regimes governed by a single dimensionless relaxation parameter. For a resonant excitation-exchange interaction, we obtain exact closed-form expressions for the Bloch-vector dynamics for both a generalized depolarizing channel and a generalized amplitude-damping channel acting as the reservoir-induced map. Using the Breuer--Laine--Piilo measure and a Choi-matrix CP-divisibility witness, we identify three distinct dynamical regimes across the parameter space: CP-divisible Markovian dynamics, CP-indivisible but P-divisible dynamics, and non-P-divisible non-Markovian dynamics. The boundaries between these regimes, and the structural differences between uniform and anisotropic environmental relaxation, are characterized numerically.