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arXiv:2011.08487

Notes on distinguishability of postselected computations

Zuzana Gavorová

Published Nov 17, 2020• 0 citations
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Abstract

The framework of postselection is becoming more and more important in various recent directions in Quantum Computation research. Postselection renders simple computational models able to perform general quantum computation. This was first observed for the linear optics model [E. Knill, R. Laflamme, G. J. Milburn, Nature 409, 46 (2001)], and has since provided us with many near-term candidates for the quantum advantage, commuting computations [M. J. Bremner, R. Jozsa, D. J. Shepherd, Proc. R. Soc. A 467, 459 (2011)] being the first. To facilitate the discussion of errors in the presence of postselection, we define and characterize trace-induced distance and diamond distance of postselected computations. We show counterexamples to simple properties that one would expect of any distance measure; the properties of convexity (when considering only the pure-state inputs would suffice), contractivity, and subadditivity of errors. On the positive side, we prove that certain weaker versions of contractivity and subadditivity and a number of other properties are preserved in the postselected setting. We achieve this via a "conversion lemma" that translates any inequality from the standard to the postselected setting.

DOI
10.48550/arxiv.2011.08487

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The framework of postselection is becoming more and more important in various recent directions in Quantum Computation research. Postselection renders simple computational models able to perform general quantum computation. This was first observed for the linear optics model [E. Knill, R. Laflamme, G. J. Milburn, Nature 409, 46 (2001)], and has since provided us with many near-term candidates for the quantum advantage, commuting computations [M. J. Bremner, R. Jozsa, D. J. Shepherd, Proc. R. Soc. A 467, 459 (2011)] being the first. To facilitate the discussion of errors in the presence of postselection, we define and characterize trace-induced distance and diamond distance of postselected computations. We show counterexamples to simple properties that one would expect of any distance measure; the properties of convexity (when considering only the pure-state inputs would suffice), contractivity, and subadditivity of errors. On the positive side, we prove that certain weaker versions of contractivity and subadditivity and a number of other properties are preserved in the postselected setting. We achieve this via a "conversion lemma" that translates any inequality from the standard to the postselected setting.
We previously suggested that photon exchange interactions could be used to produce nonlinear effects at the two-photon level, and similar effects have been experimentally observed by Resch et al. (e-print quant-ph/0306198). Here we note that photon exchange interactions are not useful for quantum information processing because they require the presence of substantial photon loss. This dependence on loss is somewhat analogous to the postselection required in the linear optics approach to quantum computing suggested by Knill, Laflamme, and Milburn [Nature (London) 409, 46 (2001)].
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.
Ever since Knill, Laflamme and Milburn [Nature (London) 409, 46 (2001)] showed that nondeterministic quantum logic operations could be performed with linear optical elements, additional photons (ancilla) and projective measurements, the idea of linear-optics quantum computation has attracted considerable interest. Our group has recently demonstrated several devices of this kind. We give an overview of recent experimental results, including the quantum parity check, the destructive controlled-NOT, and a cyclical quantum memory. The need for high-efficiency detection of single photons, and for detectors capable of distinguishing photon number will be discussed. Some experimental improvements towards meeting that need will be presented.
In hypothesis testing with quantum states, given a black box containing one of the two possible states, measurement is performed to detect in favor of one of the hypotheses. In postselected hypothesis testing, a third outcome is added, corresponding to not selecting any of the hypotheses. In postselected scenario, minimum error one-shot symmetric hypothesis testing is characterized in literature conditioned on the fact that one of the selected outcomes occur. We proceed further in this direction to give the set of all possible measurements that lead to the minimum error. We have given an arbitrary error-minimizing measurement in a parametric form. Note that not selecting any of the hypotheses decimates the quality of testing. We further give an example to show that these measurements vary in quality. There is a need to discuss the quality of postselected hypothesis testing. We then characterize the quality of postselected hypothesis testing by defining a new metric acceptance and give expression of acceptance for an arbitrary error-minimizing measurement in terms of some parameters of the measurement. On the set of measurements that achieve minimum error, we have maximized the acceptance, and given an example which achieves that, thus giving an example of the best possible measurement in terms of acceptance.
The diamond distance can measure the error between real and ideal quantum processes. We can calculate the distance for trace-preserving channels using convex optimization. However, convex optimization cannot solve the distance for non-trace-preserving channels because they are nonlinear. We generalize the diamond distance for non-trace-preserving channels. We first compute the distance for a simple measurement and post-selection channel, then generalize the method to any non-trace-preserving channels. As an example, we analyze the Knill-Laflamme-Milburn (KLM) protocol in linear optical quantum computing with physical imperfections. Our method provides a well-behaved measure for general quantum channels, which can bridge the physical imperfections with the logical errors in quantum circuits. Hence, it has broad applications in analyzing real systems in quantum information processing, such as quantum computers, quantum teleportation, measurement-based quantum computation, etc.
Space-bounded computation has been a central topic in classical and quantum complexity theory. In the quantum case, every elementary gate must be unitary. This restriction makes it unclear whether the power of space-bounded computation changes by allowing intermediate measurement. In the bounded error case, Fefferman and Remscrim [STOC 2021, pp.1343--1356] and Girish, Raz and Zhan~[ICALP 2021, pp.73:1--73:20] recently provided the break-through results that the power does not change. This paper shows that a similar result holds for space-bounded quantum computation with postselection. Namely, it is proved possible to eliminate intermediate postselections and measurements in the space-bounded quantum computation in the bounded-error setting. Our result strengthens the recent result by Le Gall, Nishimura and Yakaryilmaz~[TQC 2021, pp.10:1--10:17] that logarithmic-space bounded-error quantum computation with intermediate postselections and measurements is equivalent in computational power to logarithmic-space unbounded-error probabilistic computation. As an application, it is shown that bounded-error space-bounded one-clean qubit computation (DQC1) with postselection is equivalent in computational power to unbounded-error space-bounded probabilistic computation, and the computational supremacy of the bounded-error space-bounded DQC1 is interpreted in complexity-theoretic terms.