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arXiv:2112.14283 · Quantum

Operational Quantum Average-Case Distances

Filip B. Maciejewski, Zbigniew Puchała, Michał Oszmaniec

Published Sep 11, 2023• 8 citations
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Abstract

We introduce distance measures between quantum states, measurements, and channels based on their statistical distinguishability in generic experiments. Specifically, we analyze the average Total Variation Distance (TVD) between output statistics of protocols in which quantum objects are intertwined with random circuits and measured in standard basis. We show that for circuits forming approximate 4-designs, the average TVDs can be approximated by simple explicit functions of the underlying objects – the average-case distances (ACDs). We apply them to analyze the effects of noise in quantum advantage experiments and for efficient discrimination of high-dimensional states and channels without quantum memory. We argue that ACDs are better suited for assessing the quality of NISQ devices than common distance measures such as trace distance or the diamond norm.

DOI
10.22331/q-2023-09-11-1106

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We introduce distance measures between quantum states, measurements, and channels based on their statistical distinguishability in generic experiments. Specifically, we analyze the average Total Variation Distance (TVD) between output statistics of protocols in which quantum objects are intertwined with random circuits and measured in standard basis. We show that for circuits forming approximate 4-designs, the average TVDs can be approximated by simple explicit functions of the underlying objects – the average-case distances (ACDs). We apply them to analyze the effects of noise in quantum advantage experiments and for efficient discrimination of high-dimensional states and channels without quantum memory. We argue that ACDs are better suited for assessing the quality of NISQ devices than common distance measures such as trace distance or the diamond norm.
In this work, we present an in-depth study of average-case quantum distances introduced in Maciejewski et al. (2022). The average-case distances approximate, up to the relative error, the average Total-Variation (TV) distance between measurement outputs of two quantum processes, in which quantum objects of interest (states, measurements, or channels) are intertwined with random quantum circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantifyaverage-casestatistical distinguishability via random quantum circuits. We prove that once a family of random circuits forms an$\delta −approximate4−design,with-approximate 4-design, with−approximate4−design,with\delta =o(d^{-8}),thentheaverage−casedistancescanbeapproximatedbysimpleexplicitfunctionsthatcanbeexpressedviasimpledegreetwopolynomialsinobjectsofinterest.Forsystemsofmoderatedimension,theycanbeeasilyexplicitlycomputed–nooptimizationisneededasopposedtodiamondnormdistancebetweenchannelsoroperationaldistancebetweenmeasurements.Weprovethatthosefunctions,whichwecallquantumaverage−casedistances,haveaplethoraofdesirableproperties,suchassubadditivityw.r.t.tensorproducts,jointconvexity,and(restricted)data−processinginequalities.Notably,alldistancesutilizetheHilbert−Schmidt(HS)norm,whichprovidesthisnormwithanewoperationalinterpretation.Wealsoprovideupperboundsonthemaximalratiobetweenworst−caseandaverage−casedistances,andforeachofthem,weprovideanexamplethatsaturatesthebound.Specifically,weshowthatforeachdimension, then the average-case distances can be approximated by simple explicit functions that can be expressed via simple degree two polynomials in objects of interest. For systems of moderate dimension, they can be easily explicitly computed – no optimization is needed as opposed to diamond norm distance between channels or operational distance between measurements. We prove that those functions, which we call quantum average-case distances, have a plethora of desirable properties, such as subadditivity w.r.t. tensor products, joint convexity, and (restricted) data-processing inequalities. Notably, all distances utilize the Hilbert-Schmidt (HS) norm, which provides this norm with a new operational interpretation. We also provide upper bounds on the maximal ratio between worst-case and average-case distances, and for each of them, we provide an example that saturates the bound. Specifically, we show that for each dimension,thentheaverage−casedistancescanbeapproximatedbysimpleexplicitfunctionsthatcanbeexpressedviasimpledegreetwopolynomialsinobjectsofinterest.Forsystemsofmoderatedimension,theycanbeeasilyexplicitlycomputed–nooptimizationisneededasopposedtodiamondnormdistancebetweenchannelsoroperationaldistancebetweenmeasurements.Weprovethatthosefunctions,whichwecallquantumaverage−casedistances,haveaplethoraofdesirableproperties,suchassubadditivityw.r.t.tensorproducts,jointconvexity,and(restricted)data−processinginequalities.Notably,alldistancesutilizetheHilbert−Schmidt(HS)norm,whichprovidesthisnormwithanewoperationalinterpretation.Wealsoprovideupperboundsonthemaximalratiobetweenworst−caseandaverage−casedistances,andforeachofthem,weprovideanexamplethatsaturatesthebound.Specifically,weshowthatforeachdimensiondthisratioisatmostthis ratio is at mostthisratioisatmostd^{\frac {1}{2}}, d, d^{\frac {3}{2}}$for states, measurements, and channels, respectively. To support the practical usefulness of our findings, we study multiple examples in which average-case quantum distances can be calculated analytically.
In this article, we analyzed the performance of several classes of quantum circuits under various errors. We executed NISQ circuits on both noisy simulator and real hardware backend publicly available from IBM Quantum. On simulation, we created noise models with both individual errors and combination of multiple errors to quantify how each type of error impact performance of quantum circuits. We utilized statistical distances such as total variation distance (TVD), Jensen-Shannon divergence (JSD), and Hellinger distance, to measure the deviation of noisy output from ideal output. Finally, we compare the noisy outputs from the simulator and outputs from the real hardware to showcase the gap between simulation and real device experiment. This gap points to the limitation of present noise models failure to exhibit the behavior of real hardware.
In this article, we analyzed the performance of several classes of quantum circuits under various errors. We executed NISQ circuits on both noisy simulator and real hardware backend publicly available from IBM Quantum. On simulation, we created noise models with both individual errors and combination of multiple errors to quantify how each type of error impact performance of quantum circuits. We utilized statistical distances such as total variation distance (TVD), Jensen-Shannon divergence (JSD), and Hellinger distance, to measure the deviation of noisy output from ideal output. Finally, we compare the noisy outputs from the simulator and outputs from the real hardware to showcase the gap between simulation and real device experiment. This gap points to the limitation of present noise models failure to exhibit the behavior of real hardware.
In the Noisy Intermediate-Scale Quantum (NISQ) realm, efficient quantum circuit compilation is critical to ensure successful computation. Several third-party compilers are improving the compilation times and depth/gate counts. Untrusted third parties or a particular version of a trusted compiler may allow an attacker to steal, clone, and/or reverse engineer the quantum circuit. We propose to obfuscate quantum circuits by employing dummy CNOT gates to prevent such threats. If the adversary clones the obfuscated design, he/she will get faulty results. We propose a metric-based dummy gate insertion process to ensure maximum corruption of functionality measured using Total Variation Distance (TVD) and validated using IBM’s noisy simulators. Our metric guided dummy gate insertion process achieves TVD of up to 28.83%, and performs 10.14% better than the average TVD and performs within 12.45% of the best obtainable TVD for the test benchmarks. The removal of dummy gates by the designer post-compilation to restore functionality as well as other finer details have been addressed.
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.