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arXiv:2411.15148

Coherence in Property Testing: Quantum-Classical Collapses and Separations

Fernando Granha Jeronimo, Nir Magrafta, Joseph Slote, Pei Wu

Published Nov 6, 2024• 0 citations
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Abstract

Understanding the power and limitations of classical and quantum information and how they differ is a fundamental endeavor. In property testing of distributions, a tester is given samples over a typically large domain $\{0,1\}^n$. An important property is the support size both of distributions [Valiant and Valiant, STOC'11], as well, as of quantum states. Classically, even given $2^{n/16}$ samples, no tester can distinguish distributions of support size $2^{n/8}$ from $2^{n/4}$ with probability better than $2^{-Θ(n)}$, even promised they are flat. Quantum states can be in a coherent superposition of states of $\{0,1\}^n$, so one may ask if coherence can enhance property testing. Flat distributions naturally correspond to subset states, $|ϕ_S \rangle=1/\sqrt{|S|}\sum_{i\in S}|i\rangle$. We show that coherence alone is not enough, Coherence limitations: Given $2^{n/16}$ copies, no tester can distinguish subset states of size $2^{n/8}$ from $2^{n/4}$ with probability better than $2^{-Θ(n)}$. The hardness persists even with multiple public-coin AM provers, Classical hardness with provers: Given $2^{O(n)}$ samples from a distribution and $2^{O(n)}$ communication with AM provers, no tester can estimate the support size up to factors $2^{Ω(n)}$ with probability better than $2^{-Θ(n)}$. Our result is tight. In contrast, coherent subset state proofs suffice to improve testability exponentially, Quantum advantage with certificates: With poly-many copies and subset state proofs, a tester can approximate the support size of a subset state of arbitrary size. Some structural assumption on the quantum proofs is required since we show, Collapse of QMA: A general proof cannot improve testability of any quantum property whatsoever. We also show connections to disentangler and quantum-to-quantum transformation lower bounds.

DOI
10.48550/arxiv.2411.15148

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Understanding the power and limitations of classical and quantum information and how they differ is a fundamental endeavor. In property testing of distributions, a tester is given samples over a typically large domain {0,1}n\{0,1\}^n{0,1}n. An important property is the support size both of distributions [Valiant and Valiant, STOC'11], as well, as of quantum states. Classically, even given 2n/162^{n/16}2n/16 samples, no tester can distinguish distributions of support size 2n/82^{n/8}2n/8 from 2n/42^{n/4}2n/4 with probability better than 2−Θ(n)2^{-Θ(n)}2−Θ(n), even promised they are flat. Quantum states can be in a coherent superposition of states of {0,1}n\{0,1\}^n{0,1}n, so one may ask if coherence can enhance property testing. Flat distributions naturally correspond to subset states, ∣ϕS⟩=1/∣S∣∑i∈S∣i⟩|ϕ_S \rangle=1/\sqrt{|S|}\sum_{i\in S}|i\rangle∣ϕS​⟩=1/∣S∣​∑i∈S​∣i⟩. We show that coherence alone is not enough, Coherence limitations: Given 2n/162^{n/16}2n/16 copies, no tester can distinguish subset states of size 2n/82^{n/8}2n/8 from 2n/42^{n/4}2n/4 with probability better than 2−Θ(n)2^{-Θ(n)}2−Θ(n). The hardness persists even with multiple public-coin AM provers, Classical hardness with provers: Given 2O(n)2^{O(n)}2O(n) samples from a distribution and 2O(n)2^{O(n)}2O(n) communication with AM provers, no tester can estimate the support size up to factors 2Ω(n)2^{Ω(n)}2Ω(n) with probability better than 2−Θ(n)2^{-Θ(n)}2−Θ(n). Our result is tight. In contrast, coherent subset state proofs suffice to improve testability exponentially, Quantum advantage with certificates: With poly-many copies and subset state proofs, a tester can approximate the support size of a subset state of arbitrary size. Some structural assumption on the quantum proofs is required since we show, Collapse of QMA: A general proof cannot improve testability of any quantum property whatsoever. We also show connections to disentangler and quantum-to-quantum transformation lower bounds.
The area of property testing tries to design algorithms that can efficiently handle very large amounts of data: given a large object that either has a certain property or is somehow "far" from having that property, a tester should efficiently distinguish between these two cases. In this survey we describe recent results obtained for quantum property testing. This area naturally falls into three parts. First, we may consider quantum testers for properties of classical objects. We survey the main examples known where quantum testers can be much (sometimes exponentially) more efficient than classical testers. Second, we may consider classical testers of quantum objects. This is the situation that arises for instance when one is trying to determine if quantum states or operations do what they are supposed to do, based only on classical input-output behavior. Finally, we may also consider quantum testers for properties of quantum objects, such as states or operations. We survey known bounds on testing various natural properties, such as whether two states are equal, whether a state is separable, whether two operations commute, etc. We also highlight connections to other areas of quantum information theory and mention a number of open questions.
Recently, Doosti et al. introduced the problem of distributed quantum state verification, where mmm distributed nodes are given a copy of an unknown state ρρρ, and can send limited one way communication to a central node, who has a complete description of a known state σσσ. They ask how many distributed nodes mmm are required, before the central node can succeed at distinguishing whether ρ=σρ=σρ=σ or ∥ρ−σ∥1≥ε\|ρ-σ\|_1\geq\varepsilon∥ρ−σ∥1​≥ε with high probability. In the setting where only quantum communication is allowed, Doosti et al. exhibit conditional lower bounds in both the public and private-coin settings, and a matching upper bound in the public-coin setting. We extend these results, and show unconditional lower bounds for when both classical and quantum communication are permitted. We show the public-coin lower bound is tight by giving an algorithm with a matching upper bound. We also show an almost tight upper bound in the private-coin setting when only quantum communication is permitted.
The area of property testing tries to design algorithms that can efficiently handle very large amounts of data: given a large object that either has a certain property or is somehow “far” from having that property, a tester should efficiently distinguish between these two cases. In this survey we describe recent results obtained for quantum property testing. This area naturally falls into three parts. First, we may consider quantum testers for properties of classical objects. We survey the main examples known where quantum testers can be much (sometimes exponentially) more efficient than classical testers. Second, we may consider classical testers of quantum objects. These arise for instance when one is trying to determine if quantum states or operations do what they are supposed to do, based only on classical input-output behavior. Finally, we may also consider quantum testers for properties of quantum objects, such as states or operations. We survey known bounds on testing various natural properties, such as whether two states are equal, whether a state is separable, whether two operations commute, etc. We also highlight connections to other areas of quantum information theory and mention a number of open questions.
We study the problem of property testing quantum states: given a pure state ∣ψ⟩|ψ\rangle∣ψ⟩ that either (A) belongs to some class CCC of pure states, or (B) is ε\varepsilonε-far in trace distance from all states in CCC, determine which is the case with high probability. It is known that there exists classes CCC for which the sample complexity of property testing necessarily scales with the size of the system. In this work, we identify classes for which testing only requires a size-independent number of samples. We prove a tight sample complexity of Θ(1/ε2)Θ(1/\varepsilon^2)Θ(1/ε2) for the property testing of five important classes of Gaussian quantum states: 1) Fermionic Gaussian states, 2) Slater determinants, 3) bosonic Gaussian states, 4) zero-mean bosonic Gaussian states, and 5) bosonic coherent states. While property testers for some of these classes have been studied previously, those analyses only gave upper bounds scaling polynomially in the number of modes. In each case, our tester is the projection onto the ``top'' irrep of two or three copies. We get to constant sample complexity by lower bounding how quickly the rejection probability increases with distance from the class; the main technical component involves bounding the spectral gap of symmetric extension of the test projector. We then establish optimality by proving matching lower bounds based on binary state discrimination for each class. Our testers and optimality claims also extend to the tolerant setting, wherein ψψψ may be O(ε)O(\varepsilon)O(ε)-close to CCC in case (A), rather than lying exactly inside.
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