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arXiv:1906.09253 · Quantum

Variational Quantum Fidelity Estimation

Marco Cerezo, Alexander Poremba, Lukasz Cincio, Patrick J. Coles

Published Mar 26, 2020• 126 citations
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Abstract

Computing quantum state fidelity will be important to verify and characterize states prepared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity F ( ρ , σ ) based on the ``truncated fidelity'' F ( ρ m , σ ) , which is evaluated for a state ρ m obtained by projecting ρ onto its m -largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with m . To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that involves three steps: (1) variationally diagonalize ρ , (2) compute matrix elements of σ in the eigenbasis of ρ , and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case where σ is arbitrary and ρ is low rank, which we call low-rank fidelity estimation, and we prove that no classical algorithm can efficiently solve this problem under reasonable assumptions. Finally, we demonstrate that our bounds can detect quantum phase transitions and are often tighter than previously known computable bounds for realistic situations.

DOI
10.22331/q-2020-03-26-248

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Computing quantum state fidelity will be important to verify and characterize states prepared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity F ( ρ , σ ) based on the ``truncated fidelity'' F ( ρ m , σ ) , which is evaluated for a state ρ m obtained by projecting ρ onto its m -largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with m . To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that involves three steps: (1) variationally diagonalize ρ , (2) compute matrix elements of σ in the eigenbasis of ρ , and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case where σ is arbitrary and ρ is low rank, which we call low-rank fidelity estimation, and we prove that no classical algorithm can efficiently solve this problem under reasonable assumptions. Finally, we demonstrate that our bounds can detect quantum phase transitions and are often tighter than previously known computable bounds for realistic situations.
The quantum Fisher information (QFI) quantifies the ultimate precision of estimating a parameter from a quantum state and can be regarded as a reliability measure of a quantum system as a quantum sensor. However, estimation of the QFI for a mixed state is in general a computationally demanding task. In this paper we present a variational quantum algorithm called variational quantum Fisher information estimation (VQFIE) to address this task. By estimating lower and upper bounds on the QFI, based on bounding the fidelity, VQFIE outputs a range in which the actual QFI lies. This result can then be used to variationally prepare the state that maximizes the QFI, for the application of quantum sensing. In contrast to previous approaches, VQFIE does not require knowledge of the explicit form of the sensor dynamics. We simulate the algorithm for a magnetometry setup and demonstrate the tightening of our bounds as the state purity increases. For this example, we compare our bounds with literature bounds and show that our bounds are tighter.
Fidelity is a fundamental measure for the closeness of two quantum states, which is important both from a theoretical and a practical point of view. Yet, in general, it is difficult to give good estimates of fidelity, especially when one works with mixed states over Hilbert spaces of very high dimension. Although, there has been some progress on fidelity estimation, all prior work either requires a large number of identical copies of the relevant states, or relies on unproven heuristics. In this work, we improve on both of these aspects by developing new and efficient quantum algorithms for fidelity estimation with provable performance guarantees in case at least one of the states is approximately low-rank. Our algorithms use advanced quantum linear algebra techniques, such as the quantum singular value transformation, as well as density matrix exponentiation and quantum spectral sampling. As a complementary result, we prove that fidelity estimation to any non-trivial constant additive accuracy is hard in general, by giving a sample complexity lower bound that depends polynomially on the dimension. Moreover, if circuit descriptions for the relevant states are provided, we show that the task is hard for the complexity class called (honest verifier) quantum statistical zero knowledge via a reduction to a closely related result by Watrous.
Computation of Uhlmann fidelity between many-body mixed states generally involves full diagonalization of exponentially large matrices. In this work, we introduce a polynomial-time algorithm to compute certified lower and upper bounds for the fidelity between matrix product density operators (MPDOs). Our method maps the fidelity estimation problem to a variational optimization of sequential quantum circuits, allowing for systematic improvement of the lower bounds by increasing the circuit depth. Complementarily, we obtain certified upper bounds on fidelity by variational lower bounds on the trace distance through the same framework. We demonstrate the power of this approach with two examples: fidelity correlators in critical mixed states, and codeword distinguishability in an approximate quantum error-correcting code. Remarkably, the variational lower bound accurately track the universal scaling behavior of the fidelity with a size-consistent relative error, allowing for the extraction of previously unknown critical exponents. Our results offer an exponential improvement in precision over known moment-based bounds and establish a scalable framework for the verification of many-body quantum systems.
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.
For two unknown mixed quantum states$\rho andandand\sigma inanin aninanN−dimensionalHilbertspace,computingtheirfidelity-dimensional Hilbert space, computing their fidelity−dimensionalHilbertspace,computingtheirfidelityF(\rho,\sigma)isabasicproblemwithmanyimportantapplicationsinquantumcomputingandquantuminformation,forexampleverificationandcharacterizationoftheoutputsofaquantumcomputer,anddesignandanalysisofquantumalgorithms.Inthispaper,weproposeaquantumalgorithmthatsolvesthisprobleminis a basic problem with many important applications in quantum computing and quantum information, for example verification and characterization of the outputs of a quantum computer, and design and analysis of quantum algorithms. In this paper, we propose a quantum algorithm that solves this problem inisabasicproblemwithmanyimportantapplicationsinquantumcomputingandquantuminformation,forexampleverificationandcharacterizationoftheoutputsofaquantumcomputer,anddesignandanalysisofquantumalgorithms.Inthispaper,weproposeaquantumalgorithmthatsolvesthisproblemin{\mathrm{ poly}}(\log (N), r, 1/\varepsilon)time,wheretime, wheretime,whereristhelowerrankofis the lower rank ofisthelowerrankof\rho andandand\sigma ,and, and,and\varepsilon isthedesiredprecision,providedthatthepurificationsofis the desired precision, provided that the purifications ofisthedesiredprecision,providedthatthepurificationsof\rho andandand\sigma arepreparedbyquantumoracles.Thisalgorithmexhibitsanexponentialspeedupoverthebestknownalgorithm(basedonquantumstatetomography)whichhastimecomplexitypolynomialinare prepared by quantum oracles. This algorithm exhibits an exponential speedup over the best known algorithm (based on quantum state tomography) which has time complexity polynomial inarepreparedbyquantumoracles.Thisalgorithmexhibitsanexponentialspeedupoverthebestknownalgorithm(basedonquantumstatetomography)whichhastimecomplexitypolynomialinN$.
We study the number of copies needed to estimate the root Uhlmann fidelity between an unknown quantum state and a classically known reference, under collective measurements. If the unknown state has rank at most sss, we give an estimator using O(s2/ε2)O(s^2/\varepsilon^2)O(s2/ε2) copies, uniformly in the ambient dimension and reference rank. The estimator applies a random-purification channel and a covariant pure-state measurement, then rescales the observed amplitude before evaluating a weighted nuclear norm. Combined with the lower bound established independently in our first version and in concurrent work of Wang, this determines the worst-case complexity under rank bounds r,sr,sr,s as min⁡{r,s}2/ε2\min\{r,s\}^2/\varepsilon^2min{r,s}2/ε2 up to logarithmic factors in the smaller rank. The same estimator gives the upper bound O(r(tr⁡σ)2/ε2)O(r(\operatorname{tr}\sqrtσ)^2/\varepsilon^2)O(r(trσ​)2/ε2) for a rank-rrr reference σσσ. We combine spectral truncation with lower bounds obtained by embedding hard instances for the maximally mixed reference into spectral subspaces. For spectra λi∝i−αλ_i\propto i^{-α}λi​∝i−α with fixed 1<α≤21<α\le21<α≤2, as ε↓0\varepsilon\downarrow0ε↓0 with r≥Cαε−2/(α−1)r\ge C_α\varepsilon^{-2/(α-1)}r≥Cα​ε−2/(α−1), the bounds determine the complexity as Θ~α(ε−4/(α−1))\widetildeΘ_α(\varepsilon^{-4/(α-1)})Θα​(ε−4/(α−1)). In particular, inverse-square spectra have accuracy exponent four. The lower-bound construction uses exact moment matching and an explicitly computable Schur measure.