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arXiv:1205.0828

Property Testing of Quantum Measurements

Guoming Wang

Published May 3, 2012• 2 citations
arXivHTMLPDF

Abstract

In this paper, we study the following question: given a black box performing some unknown quantum measurement on a multi-qudit system, how do we test whether this measurement has certain property or is far away from having this property. We call this task \textit{property testing} of quantum measurement. We first introduce a metric for quantum measurements, and show that it possesses many nice features. Then we show that, with respect to this metric, the following classes of measurements can be efficiently tested: 1. the stabilizer measurements, which play a crucial role for quantum error correction; 2. the $k$-local measurements, i.e. measurements whose outcomes depend on a subsystem of at most $k$ qudits; 3. the permutation-invariant measurements, which include those measurements used in quantum data compression, state estimation and entanglement concentration. In fact, all of them can be tested with query complexity independent of the system's dimension. Furthermore, we also present an algorithm that can test any finite set of measurements. Finally, we consider the following natural question: given two black-box measurement devices, how do we estimate their distance? We give an efficient algorithm for this task, and its query complexity is also independent of the system's dimension. As a consequence, we can easily test whether two unknown measurements are identical or very different.

DOI
10.48550/arxiv.1205.0828

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In this paper, we study the following question: given a black box performing some unknown quantum measurement on a multi-qudit system, how do we test whether this measurement has certain property or is far away from having this property. We call this task \textit{property testing} of quantum measurement. We first introduce a metric for quantum measurements, and show that it possesses many nice features. Then we show that, with respect to this metric, the following classes of measurements can be efficiently tested: 1. the stabilizer measurements, which play a crucial role for quantum error correction; 2. the kkk-local measurements, i.e. measurements whose outcomes depend on a subsystem of at most kkk qudits; 3. the permutation-invariant measurements, which include those measurements used in quantum data compression, state estimation and entanglement concentration. In fact, all of them can be tested with query complexity independent of the system's dimension. Furthermore, we also present an algorithm that can test any finite set of measurements. Finally, we consider the following natural question: given two black-box measurement devices, how do we estimate their distance? We give an efficient algorithm for this task, and its query complexity is also independent of the system's dimension. As a consequence, we can easily test whether two unknown measurements are identical or very different.
Suppose one has access to oracles generating samples from two unknown probability distributions p and q on some N-element set. How many samples does one need to test whether the two distributions are close or far from each other in the L1-norm? This and related questions have been extensively studied during the last years in the field of property testing. In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the L1-distance ∥p-q∥1can be estimated with a constant precision using only O(N1/2) queries in the quantum settings, whereas classical computers need Ω(N1-o(1)) queries. We also describe quantum algorithms for testing uniformity and orthogonality with query complexity O(N1/3). The classical query complexity of these problems is known to be Ω(N1/2).
We present several new examples of speed-ups obtainable by quantum algorithms in the context of property testing. First, motivated by sampling algorithms, we consider probability distributions given in the form of an oracle f:[n]→[m]f:[n]\to[m]f:[n]→[m]. Here the probability $\PP_f(j)$ of an outcome j∈[m]j\in[m]j∈[m] is the fraction of its domain that fff maps to jjj. We give quantum algorithms for testing whether two such distributions are identical or εεε-far in L1L_1L1​-norm. Recently, Bravyi, Hassidim, and Harrow \cite{BHH10} showed that if $\PP_f$ and $\PP_g$ are both unknown (i.e., given by oracles fff and ggg), then this testing can be done in roughly m\sqrt{m}m​ quantum queries to the functions. We consider the case where the second distribution is known, and show that testing can be done with roughly m1/3m^{1/3}m1/3 quantum queries, which we prove to be essentially optimal. In contrast, it is known that classical testing algorithms need about m2/3m^{2/3}m2/3 queries in the unknown-unknown case and about m\sqrt{m}m​ queries in the known-unknown case. Based on this result, we also reduce the query complexity of graph isomorphism testers with quantum oracle access. While those examples provide polynomial quantum speed-ups, our third example gives a much larger improvement (constant quantum queries vs polynomial classical queries) for the problem of testing periodicity, based on Shor's algorithm and a modification of a classical lower bound by Lachish and Newman \cite{lachish&newman:periodicity}. This provides an alternative to a recent constant-vs-polynomial speed-up due to Aaronson \cite{aaronson:bqpph}.
We present several new examples of speed-ups obtainable by quantum algorithms in the context of property testing. First, motivated by sampling algorithms, we consider probability distributions given in the form of an oracle f:[n]→[m]f:[n]\to[m]f:[n]→[m]. Here the probability Pf(j)P_f(j)Pf​(j) of an outcome jjj in [m][m][m] is the fraction of its domain that fff maps to jjj. We give quantum algorithms for testing whether two such distributions are identical or epsilonepsilonepsilon-far in L1L_1L1​-norm. Recently, Bravyi, Hassidim, and Harrow showed that if PfP_fPf​ and PgP_gPg​ are both unknown (i.e., given by oracles fff and ggg), then this testing can be done in roughly sqrtmsqrt{m}sqrtm quantum queries to the functions. We consider the case where the second distribution is known, and show that testing can be done with roughly m1/3m^{1/3}m1/3 quantum queries, which we prove to be essentially optimal. In contrast, it is known that classical testing algorithms need about m2/3m^{2/3}m2/3 queries in the unknown-unknown case and about sqrtmsqrt{m}sqrtm queries in the known-unknown case. Based on this result, we also reduce the query complexity of graph isomorphism testers with quantum oracle access. While those examples provide polynomial quantum speed-ups, our third example gives a much larger improvement (constant quantum queries vs polynomial classical queries) for the problem of testing periodicity, based on Shor's algorithm and a modification of a classical lower bound by Lachish and Newman. This provides an alternative to a recent constant-vs-polynomial speed-up due to Aaronson.
We present fast algorithms $\unicode{x2013}$ under the depolarizing noise assumption, often made in fault-tolerant quantum computations $\unicode{x2013}$ to test its strength. Our optimal algorithms answer the following question: is the quantum channel implemented by a given black box identical to a target unitary or ε\varepsilonε-far from it in the diamond distance, assuming that the deviation is a depolarizing channel with unknown parameter? Our algorithm has a query complexity of Θ(1/ε).Θ(1/\varepsilon).Θ(1/ε). The query complexity of the relaxed problem of testing whether the black-box channel is ε1\varepsilon_1ε1​-close to a target unitary or ε2\varepsilon_2ε2​-far in the diamond distance is Θ(ε2/(ε2−ε1)2).Θ\bigl(\varepsilon_2/(\varepsilon_2 - \varepsilon_1)^2\bigr).Θ(ε2​/(ε2​−ε1​)2). In both cases, we provide matching lower bounds that hold even for adaptive, ancilla-assisted protocols with multi-outcome incoherent measurements.
We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between QMA\mathsf{QMA}QMA and QMA(2)\mathsf{QMA(2)}QMA(2), a long standing question in quantum complexity theory.
We describe two procedures which, given access to one copy of a quantum state and a sequence of two-outcome measurements, can distinguish between the case that at least one of the measurements accepts the state with high probability, and the case that all of the measurements have low probability of acceptance. The measurements cannot simply be tried in sequence, because early measurements may disturb the state being tested. One procedure is based on a variant of Marriott-Watrous amplification. The other procedure is based on the use of a test for this disturbance, which is applied with low probability. We find a number of applications: Quantum query complexity separations in the property testing model for testing isomorphism of functions under group actions. We give quantum algorithms for testing isomorphism, linear isomorphism and affine isomorphism of boolean functions which use exponentially fewer queries than is possible classically, and a quantum algorithm for testing graph isomorphism which uses polynomially fewer queries than the best algorithm known. Testing properties of quantum states and operations. We show that any finite property of quantum states can be tested using a number of copies of the state which is logarithmic in the size of the property, and give a test for genuine multipartite entanglement of states of n qubits that uses O(n) copies of the state. We also show that equivalence of two unitary operations under conjugation by a unitary picked from a fixed set can be tested efficiently. This is a natural quantum generalisation of testing isomorphism of boolean functions. Correcting an error in a result of Aaronson on de- Merlinizing quantum protocols. This result claimed that, in any one-way quantum communication protocol where two parties are assisted by an all-powerful but untrusted third party, the third party can be removed with only a modest increase in the communication cost. We give a corrected proof of a key technical lemma required for Aaronson's result.