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arXiv:2108.08406 · Physical Review A

Estimating distinguishability measures on quantum computers

Soorya Rethinasamy, Rochisha Agarwal, Kunal Sharma, Mark M. Wilde

Published Jul 10, 2023• 30 citations
arXivHTMLPDF

Abstract

The performance of a quantum information processing protocol is ultimately judged by distinguishability measures that quantify how distinguishable the actual result of the protocol is from the ideal case. The most prominent distinguishability measures are those based on the fidelity and trace distance, due to their physical interpretations. In this paper we propose and review several algorithms for estimating distinguishability measures based on trace distance and fidelity. The algorithms can be used for distinguishing quantum states, channels, and strategies (the last also known in the literature as ``quantum combs''). The fidelity-based algorithms offer physical interpretations of these distinguishability measures in terms of the maximum probability with which a single prover (or competing provers) can convince a verifier to accept the outcome of an associated computation. We simulate many of these algorithms by using a variational approach with parameterized quantum circuits. We find that the simulations converge well in both the noiseless and noisy scenarios for all examples considered. Furthermore, the noisy simulations exhibit a parameter noise resilience. Finally, we establish a strong relationship between various quantum computational complexity classes and distance estimation problems.

DOI
10.1103/physreva.108.012409

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The performance of a quantum information processing protocol is ultimately judged by distinguishability measures that quantify how distinguishable the actual result of the protocol is from the ideal case. The most prominent distinguishability measures are those based on the fidelity and trace distance, due to their physical interpretations. In this paper we propose and review several algorithms for estimating distinguishability measures based on trace distance and fidelity. The algorithms can be used for distinguishing quantum states, channels, and strategies (the last also known in the literature as ``quantum combs''). The fidelity-based algorithms offer physical interpretations of these distinguishability measures in terms of the maximum probability with which a single prover (or competing provers) can convince a verifier to accept the outcome of an associated computation. We simulate many of these algorithms by using a variational approach with parameterized quantum circuits. We find that the simulations converge well in both the noiseless and noisy scenarios for all examples considered. Furthermore, the noisy simulations exhibit a parameter noise resilience. Finally, we establish a strong relationship between various quantum computational complexity classes and distance estimation problems.
Abstract Estimating the difference between quantum data is crucial in quantum computing. However, as typical characterizations of quantum data similarity, the trace distance and quantum fidelity are believed to be exponentially-hard to evaluate in general. In this work, we introduce hybrid quantum–classical algorithms for these two distance measures on near-term quantum devices where no assumption of input state is required. First, we introduce the variational trace distance estimation (VTDE) algorithm. We in particular provide the technique to extract the desired spectrum information of any Hermitian matrix by local measurement. A novel variational algorithm for trace distance estimation is then derived from this technique, with the assistance of a single ancillary qubit. Notably, VTDE could avoid the barren plateau issue with logarithmic depth circuits due to a local cost function. Second, we introduce the variational fidelity estimation algorithm. We combine Uhlmann’s theorem and the freedom in purification to translate the estimation task into an optimization problem over a unitary on an ancillary system with fixed purified inputs. We then provide a purification subroutine to complete the translation. Both algorithms are verified by numerical simulations and experimental implementations, exhibiting high accuracy for randomly generated mixed states.
Measuring the distinguishability between quantum states is a basic problem in quantum information theory. In this paper, we develop optimal quantum algorithms that estimate both the trace distance and the (square root) fidelity between pure states to within additive error$\varepsilon usingusingusing\Theta (1/\varepsilon)queriestotheirstate−preparationcircuits,quadraticallyimprovingthelong−standingfolklorequeries to their state-preparation circuits, quadratically improving the long-standing folklorequeriestotheirstate−preparationcircuits,quadraticallyimprovingthelong−standingfolkloreO(1/\varepsilon ^{2}) $. At the heart of our construction, is an algorithmic tool for quantum square root amplitude estimation, which generalizes the well-known quantum amplitude estimation.
The data-processing inequality ensures quantum channels reduce state distinguishability, with contraction coefficients quantifying optimal bounds. However, these can be overly optimistic and not representative of the usual behavior. We study how noise contracts distinguishability of `typical' states, beyond the worst-case. To that end, we introduce and study a family of moments of contraction for quantum divergences, which interpolate between the worst-case contraction coefficient of a channel and its average behavior under a chosen ensemble of input states. We establish general properties of these moments, relate moments for different divergences, and derive bounds in terms of channel parameters like the entropy or purity of its Choi state. Focusing on the trace distance, we obtain upper and lower bounds on its average contraction under tensor-product noise channels, and prove that, depending on the local noise strength, there is a phase transition in the limit of many channel uses: below a critical error rate the average contraction remains near unity, whereas above it decays exponentially with system size. We extend these phase-transition phenomena to random quantum circuits with unital noise, showing that constant-depth noisy circuits do not shrink the trace distance on average, even when given highly entangled states as input. In contrast, even at log⁡log⁡n\log\log nloglogn depth, the average trace distance can become superpolynomially small. Finally, we explore moments of contraction for f-divergences and discuss applications to local differential privacy, demonstrating that noise regimes ensuring privacy can render outputs essentially indistinguishable on average. Thus, our results provide a fine-grained framework to quantify typical channel noise in quantum information and computation and unveil new phenomena in contraction coefficients, such as phase transitions for average contraction.
The fidelity function for quantum states has been widely used in quantum information science and frequently arises in the quantification of optimal performances for the estimation and distinguishing of quantum states. A fidelity function for quantum channels is expected to have the same wide applications in quantum information science. In this paper we propose a fidelity function for quantum channels and show that various distance measures on quantum channels can be obtained from this fidelity function; for example, the Bures angle and the Bures distance can be extended to quantum channels via this fidelity function. We then show that the distances between quantum channels lead naturally to a quantum channel Fisher information which quantifies the ultimate precision limit in quantum metrology; the ultimate precision limit can thus be seen as a manifestation of the distances between quantum channels. We also show that the fidelity of quantum channels provides a unified framework for perfect quantum channel discrimination and quantum metrology. In particular, we show that the minimum number of uses needed for perfect channel discrimination is exactly the counterpart of the precision limit in quantum metrology, and various useful lower bounds for the minimum number of uses needed for perfect channel discrimination can be obtained via this connection.
The characterization of noisy quantum circuits is an important step in the development of large-scale quantum computers. As experimental quantum architectures approach the threshold for fault-tolerant quantum computing, obtaining a benchmark of process fidelities is needed to verify the integrity of the system and ensure functionality. Additionally, the ability to successfully implement a quantum code of choice may depend on the type of noise that is present in the system. It is therefore necessary to characterize not only the fidelity of the process, but also the structure of the noise. \n \nRandomized benchmarking is a characterization protocol which extracts partial information from a noisy quantum process. This thesis gives a detailed review of the randomized benchmarking protocol, and presents the results of three collaborative projects which use techniques similar to randomized benchmarking to obtain previously hidden information. The first protocol is designed to obtain a benchmark on loss errors. This protocol also provides information on the detector efficiency and an additional constraint on estimated parameters from randomized benchmarking. The second protocol is an extension of the first, under the special case of coherent leakage errors, for which we obtain an estimate of the leakage rate. The third protocol provides a method of testing for spatial correlations in noisy processes. This test may be used to determine the correlation structure of errors, and the error correcting codes needed for successful experiments in a particular architecture. All protocols presented in this thesis are scalable, platform-independent and insensitive to state preparation and measurement errors. We also provide results of numerical tests of these protocols for simulated noise, which show their accuracy and robustness.
Quantum fidelity estimation is essential for benchmarking quantum states and processes on noisy quantum devices. While stabilizer operations form the foundation of fault-tolerant quantum computing, nonstabilizer resources further enable universal quantum computation through state injection. In this work, we propose several efficient quantum fidelity estimation protocols for both quantum states and channels within the resource theory of nonstabilizerness, focusing on qudit systems with odd prime dimensions. Our fidelity estimation algorithms require measuring only a constant number of phase-space point operator expectation values, with operators selected randomly according to an importance weighting scheme tailored to the target state. Notably, we demonstrate that mathematically defined nonstabilizerness measures---such as Wigner rank and mana---quantify the sample complexity of the proposed protocols, thereby endowing them with a clear operational interpretation in the fidelity estimation task. This connection reveals a fundamental trade-off: while fidelity estimation for general quantum states and channels requires resources that scale exponentially with their nonstabilizerness, the task remains tractable for states and channels that admit efficient classical simulation.