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arXiv:2204.07155 · Annual Symposium on Foundations of Computer Science

Tight Bounds for Quantum State Certification with Incoherent Measurements

Sitan Chen, Jerry Zheng Li, Brice Huang, Allen Liu

Published Oct 1, 2022• 21 citations
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Abstract

We consider the problem of quantum state certification, where we are given the description of a mixed state $\sigma\in\mathbb{C}^{d\times d},n$ copies of a mixed state $\rho\in\mathbb{C}^{d\times d}$, and $\varepsilon\gt 0$, and we are asked to determine whether $\rho=\sigma$ or whether $\|\rho-\sigma\|_{1}\gt \varepsilon$. When $\sigma$ is the maximally mixed state $\frac{1}{d}I_{d}$, this is known as mixedness testing. We focus on algorithms which use incoherent measurements, i.e. which only measure one copy of $\rho$ at a time. Unlike those that use entangled, multi-copy measurements, these can be implemented without persistent quantum memory and thus represent a large class of protocols that can be run on current or near-term devices. For mixedness testing, there is a folklore algorithm which uses incoherent measurements and only needs $O(d^{3/2}/\varepsilon^{2})$ copies. The algorithm is non-adaptive, that is, its measurements are fixed ahead of time, and is known to be optimal for non-adaptive algorithms. However, when the algorithm can make arbitrary incoherent measurements, the best known lower bound is only $\Omega(d^{4/3}/\varepsilon^{2})$ [5], and it has been an outstanding open problem to close this polynomial gap. In this work: •We settle the copy complexity of mixedness testing with incoherent measurements and show that $\Omega(d^{3/2}/\varepsilon^{2})$ copies are necessary. This fully resolves open questions of [15] and [5].•We show that the instance-optimal bounds for state certification to general $\sigma$ first derived in [7] for non-adaptive measurements also hold for arbitrary incoherent measurements. Qualitatively, our results say that adaptivity does not help at all for these problems. Our results are based on new techniques that allow us to reduce the problem to understanding the concentration of certain matrix martingales, which we believe may be of independent interest.

DOI
10.1109/focs54457.2022.00118

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We consider the problem of quantum state certification, where we are given the description of a mixed state σ∈Cd×d,n\sigma\in\mathbb{C}^{d\times d},nσ∈Cd×d,n copies of a mixed state ρ∈Cd×d\rho\in\mathbb{C}^{d\times d}ρ∈Cd×d, and ε>0\varepsilon\gt 0ε>0, and we are asked to determine whether ρ=σ\rho=\sigmaρ=σ or whether ∥ρ−σ∥1>ε\|\rho-\sigma\|_{1}\gt \varepsilon∥ρ−σ∥1​>ε. When σ\sigmaσ is the maximally mixed state 1dId\frac{1}{d}I_{d}d1​Id​, this is known as mixedness testing. We focus on algorithms which use incoherent measurements, i.e. which only measure one copy of ρ\rhoρ at a time. Unlike those that use entangled, multi-copy measurements, these can be implemented without persistent quantum memory and thus represent a large class of protocols that can be run on current or near-term devices. For mixedness testing, there is a folklore algorithm which uses incoherent measurements and only needs O(d3/2/ε2)O(d^{3/2}/\varepsilon^{2})O(d3/2/ε2) copies. The algorithm is non-adaptive, that is, its measurements are fixed ahead of time, and is known to be optimal for non-adaptive algorithms. However, when the algorithm can make arbitrary incoherent measurements, the best known lower bound is only Ω(d4/3/ε2)\Omega(d^{4/3}/\varepsilon^{2})Ω(d4/3/ε2) [5], and it has been an outstanding open problem to close this polynomial gap. In this work: •We settle the copy complexity of mixedness testing with incoherent measurements and show that Ω(d3/2/ε2)\Omega(d^{3/2}/\varepsilon^{2})Ω(d3/2/ε2) copies are necessary. This fully resolves open questions of [15] and [5].•We show that the instance-optimal bounds for state certification to general σ\sigmaσ first derived in [7] for non-adaptive measurements also hold for arbitrary incoherent measurements. Qualitatively, our results say that adaptivity does not help at all for these problems. Our results are based on new techniques that allow us to reduce the problem to understanding the concentration of certain matrix martingales, which we believe may be of independent interest.
In this work, we consider the fundamental task of quantum state certification: given copies of an unknown quantum state ρρρ, test whether it matches some target state σσσ or is εεε-far from it. For certifying ddd-dimensional states, Θ(d/ε2)Θ(d/ε^2)Θ(d/ε2) copies of ρρρ are known to be necessary and sufficient. However, the algorithm achieving this complexity makes fully entangled measurements over all O(d/ε2)O(d/ε^2)O(d/ε2) copies of ρρρ. Often, one is interested in certifying states to a high precision; this makes such joint measurements intractable even for low-dimensional states. Thus, we study whether one can obtain optimal rates for quantum state certification and related testing problems while only performing measurements on ttt copies at once, for some 1<t≪d/ε21 < t \ll d/ε^21<t≪d/ε2. While it is well-understood how to use intermediate entanglement to achieve optimal quantum state learning, the only protocol known to achieve optimal testing is the one using fully entangled measurements. Our main result is a smooth copy complexity upper bound for state certification as a function of ttt, which achieves a near-optimal rate at t=d2t = d^2t=d2. In the high-precision regime, i.e., for ε<1dε< \frac{1}{\sqrt{d}}ε<d​1​, this is a strict improvement over the entanglement used by the aforementioned optimal protocol. We also extend our techniques to develop new algorithms for the related tasks of mixedness testing and purity estimation, and show tradeoffs achieving the optimal rates for these problems at t=d2t = d^2t=d2 as well. Our algorithms are based on novel reductions from testing to learning and leverage recent advances in quantum state tomography in a non-black-box fashion. We complement our testing upper bounds with a (nearly) matching lower bound that holds in the high-precision regime, implying that one must take t≥Ω(d2)t \geq Ω(d^2)t≥Ω(d2) to achieve the optimal rate for certification.
We revisit the basic problem of quantum state certification: given copies of unknown mixed state ρ∈Cd×dρ\in\mathbb{C}^{d\times d}ρ∈Cd×d and the description of a mixed state σσσ, decide whether σ=ρσ= ρσ=ρ or ∥σ−ρ∥tr≥ε\|σ- ρ\|_{\mathsf{tr}} \ge ε∥σ−ρ∥tr​≥ε. When σσσ is maximally mixed, this is mixedness testing, and it is known that Ω(dΘ(1)/ε2)Ω(d^{Θ(1)}/ε^2)Ω(dΘ(1)/ε2) copies are necessary, where the exact exponent depends on the type of measurements the learner can make [OW15, BCL20], and in many of these settings there is a matching upper bound [OW15, BOW19, BCL20]. Can one avoid this dΘ(1)d^{Θ(1)}dΘ(1) dependence for certain kinds of mixed states σσσ, e.g. ones which are approximately low rank? More ambitiously, does there exist a simple functional f:Cd×d→R≥0f:\mathbb{C}^{d\times d}\to\mathbb{R}_{\ge 0}f:Cd×d→R≥0​ for which one can show that Θ(f(σ)/ε2)Θ(f(σ)/ε^2)Θ(f(σ)/ε2) copies are necessary and sufficient for state certification with respect to any σσσ? Such instance-optimal bounds are known in the context of classical distribution testing, e.g. [VV17]. Here we give the first bounds of this nature for the quantum setting, showing (up to log factors) that the copy complexity for state certification using nonadaptive incoherent measurements is essentially given by the copy complexity for mixedness testing times the fidelity between σσσ and the maximally mixed state. Surprisingly, our bound differs substantially from instance optimal bounds for the classical problem, demonstrating a qualitative difference between the two settings.
We consider a fundamental problem of \emph{mixedness testing}: Given nnn copies of an NNN-qubit state ρρρ, determine whether ρ=Id/dρ= \mathbb{I}_d/dρ=Id​/d or ∥ρ−Id/d∥1≥ε\|ρ-\mathbb{I}_d/d\|_1 \geq \varepsilon∥ρ−Id​/d∥1​≥ε with high probability, where d=2Nd = 2^Nd=2N. In particular, we focus on performing this task in the practical setting of single-qubit measurements, where measurements are prepared independently on each qubit. We provide a nearly complete picture of single-qubit mixedness tesing by showing n=Θ~(10N/ε2)n = \tildeΘ\left(\sqrt{10}^N/\varepsilon^2\right)n=Θ~(10​N/ε2). To establish our lower bound, we introduce a measurement-dependent lower bound framework for adaptive single-copy state certification. For the upper bound, we present a randomized Pauli basis measurement protocol, which relies on a new primitive for computationally efficient uniformity testing of correlation-concentrated distributions on the Boolean hypercube. In conjunction, we provide lower and upper bounds for mixedness testing with fixed Pauli measurement protocols.
In this work, we consider the fundamental task of estimating the purity of an unknown state to within multiplicative\textit{multiplicative}multiplicative error ε\varepsilonε. When one can perform general collective measurements, Θ(dε2+dε)Θ\left(\frac{\sqrt{d}}{\varepsilon^2} + \frac{d}{\varepsilon}\right)Θ(ε2d​​+εd​) copies are known to be necessary and sufficient for this problem [AISW20]. However, implementing collective measurements on such a large number of copies can be experimentally demanding, and we thus aim to characterize the copy complexity of this problem with incoherent measurements. In this setting, the only non-trivial result is a non-adaptive algorithm that uses O(dε2+d2ε)O\left( \frac{d}{\varepsilon^2} + \frac{d^2}{\varepsilon} \right)O(ε2d​+εd2​) copies [PTTW26], which is polynomially larger than the best-known lower bound. Our first result is a new algorithm performing non-adaptive incoherent measurements that succeeds using O(dε2+d3/2ε)O\left( \frac{d}{\varepsilon^2} + \frac{d^{3/2}}{\varepsilon} \right)O(ε2d​+εd3/2​) copies, improving on the latter term in the copy complexity. Moreover, we show that for any algorithm restricted to non-adaptive measurements, the above copy complexity is optimal. We also develop a new adaptive estimator for the purity of a state that improves on the above complexity in the high-precision regime, i.e., for ε=o(1/d)\varepsilon = o(1/d)ε=o(1/d). We also show that the non-adaptive and adaptive estimators together yield the optimal complexity for incoherent purity estimation; in particular, we show that the copy complexity of this problem is Θ(min⁡{dε2+d3/2ε,d2ε+dε2}). Θ\left(\min\left\{ \frac{d}{\varepsilon^2} + \frac{d^{3/2}}{\varepsilon}, \frac{d^2}{\varepsilon} + \frac{\sqrt{d}}{\varepsilon^2}\right\} \right). Θ(min{ε2d​+εd3/2​,εd2​+ε2d​​}).
We consider the classic question of state tomography: given copies of an unknown quantum state ρ∈Cd×d\rho \in \mathbb{C}^{d \times d}ρ∈Cd×d, output ρ^\widehat{\rho}ρ​ which is close to ρ\rhoρ in some sense, e.g. trace distance or fidelity. When one is allowed to make coherent measurements entangled across all copies, Θ(d2/ε2)\Theta\left(d^{2} / \varepsilon^{2}\right)Θ(d2/ε2) copies are necessary and sufficient to get trace distance ε\varepsilonε [18], [29]. Unfortunately, the protocols achieving this rate incur large quantum memory overheads that preclude implementation on near-term devices. On the other hand, the best known protocol using incoherent (single-copy) measurements uses O(d3/ε2)O\left(d^{3} / \varepsilon^{2}\right)O(d3/ε2) copies [24], and multiple papers have posed it as an open question to understand whether or not this rate is tight [6], [18]. In this work, we fully resolve this question, by showing that any protocol using incoherent measurements, even if they are chosen adaptively, requires Ω(d3/ε2)\Omega\left(d^{3} / \varepsilon^{2}\right)Ω(d3/ε2) copies, matching the upper bound of [24]. We do so by a new proof technique which directly bounds the “tilt” of the posterior distribution after measurements, which yields a surprisingly short proof of our lower bound, and which we believe may be of independent interest. While this implies that adaptivity does not help for tomography with respect to trace distance, we show that it actually does help for tomography with respect to infidelity. We give an adaptive algorithm that outputs a state which is γ\gammaγ-close in infidelity to ρ\rhoρ using only O~(d3/γ)\widetilde{O}\left(d^{3} / \gamma\right)O(d3/γ) copies, which is optimal for incoherent measurements. In contrast, it is known [18] that any nonadaptive algorithm requires Ω(d3/γ2)\Omega\left(d^{3} / \gamma^{2}\right)Ω(d3/γ2) copies. While it is folklore that in 2 dimensions, one can achieve a scaling of O(1/γ)O(1 / \gamma)O(1/γ), to the best of our knowledge, our algorithm is the first to achieve the optimal rate in all dimensions.
We study the problems of quantum state certification, equivalence testing and independence testing. In certification, given samples of an unknown quantum state ρρρ and the description of a state σσσ, the goal is to test whether ρ=σρ=σρ=σ, or whether ρρρ and σσσ are far in a given distance measure. In equivalence testing, σσσ is also unknown and only accessible via samples. Independence testing decides whether ρAC=ρA⊗ρCρ_{AC}=ρ_A\otimesρ_CρAC​=ρA​⊗ρC​, or is far from being a product. The sample complexities of these problems are now well-understood for a decision gap ε\varepsilonε in trace distance: in the single-copy measurement setting with ddd-dimensional states, all three tasks can be solved using the same non-adaptive approach, which uses Θ(d3/2/ε2)Θ(d^{3/2}/\varepsilon^2)Θ(d3/2/ε2) samples and is optimal in general, even without adaptivity. In this work, we consider decision gaps expressed in fidelity and study possible separations between these problems and how adaptivity can help. We prove that certification with respect to fidelity for a state σσσ of rank rrr does not benefit from adaptivity and requires Θ~(r3/2/ε)\widetildeΘ(r^{3/2}/\varepsilon)Θ(r3/2/ε) samples. For equivalence testing and independence testing, we provide adaptive algorithms using O~(min⁡{d3/2/ε2,d9/4/ε})\widetilde{O}(\min\{d^{3/2}/\varepsilon^2,d^{9/4}/\varepsilon\})O(min{d3/2/ε2,d9/4/ε}) and O~(min⁡{(dAdC)3/2/ε2,dA9/4dC3/4/ε})\widetilde{O}(\min\{(d_Ad_C)^{3/2}/\varepsilon^2,d_A^{9/4}d_C^{3/4}/\varepsilon\})O(min{(dA​dC​)3/2/ε2,dA9/4​dC3/4​/ε}) samples, for dA≥dCd_A\geq d_CdA​≥dC​, respectively. Our main technique is a framework that uses partial learning and a reduction to testing in ℓ2\ell_2ℓ2​-distance, adapted from the distribution testing literature. We show that adaptivity matters for equivalence testing in fidelity by proving that Ω~(1/ε2)\widetildeΩ(1/\varepsilon^2)Ω(1/ε2) samples are necessary in the non-adaptive case even for qubits, showing a separation from certification.