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arXiv:1711.01053 · SIAM Journal on Computing

Shadow Tomography of Quantum States

Scott Aaronson

Published Jun 20, 2018• 323 citations
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Abstract

We introduce the problem of shadow tomography: given an unknown $D$-dimensional quantum mixed state $\rho$, as well as known two-outcome measurements $E_{1},\ldots,E_{M}$, estimate the probability that $E_{i}$ accepts $\rho$, to within additive error $\varepsilon$, for each of the $M$ measurements. How many copies of $\rho$ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only $\widetilde{O}\left(\varepsilon^{-4}\cdot\log^{4}M\cdot\log D\right)$ copies. This means, for example, that we can learn the behavior of an arbitrary $n$-qubit state, on all accepting/rejecting circuits of some fixed polynomial size, by measuring only $n^{O(1)}$ copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, Branda͂o et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.

DOI
10.1137/18m120275x

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We introduce the problem of shadow tomography: given an unknown DDD-dimensional quantum mixed state ρ\rhoρ, as well as known two-outcome measurements E1,…,EME_{1},\ldots,E_{M}E1​,…,EM​, estimate the probability that EiE_{i}Ei​ accepts ρ\rhoρ, to within additive error ε\varepsilonε, for each of the MMM measurements. How many copies of ρ\rhoρ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only O~(ε−4⋅log⁡4M⋅log⁡D)\widetilde{O}\left(\varepsilon^{-4}\cdot\log^{4}M\cdot\log D\right)O(ε−4⋅log4M⋅logD) copies. This means, for example, that we can learn the behavior of an arbitrary nnn-qubit state, on all accepting/rejecting circuits of some fixed polynomial size, by measuring only nO(1)n^{O(1)}nO(1) copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, Branda͂o et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.
Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown ddd-dimensional quantum state ρρρ and a known collection of observables E1,…,EME_1,\ldots,E_ME1​,…,EM​, the goal is to estimate all expectation values {Tr(ρEi)}i=1M\{\text{Tr}(ρE_i)\}_{i=1}^M{Tr(ρEi​)}i=1M​ to additive accuracy ε\varepsilonε with probability at least 1−δ1-δ1−δ. An elusive open question from the seminal shadow tomography work of Aaronson is whether this task admits a dimension-independent sample complexity with only polylogarithmic dependence on MMM, as suggested by the best-known lower bounds. In this work, we propose two different quantum protocols for shadow tomography with the best sample complexity O(log⁡(M)log⁡(M/δ)ε2), O\left( \frac{\log(M)\log(M/δ)}{\varepsilon^2} \right), O(ε2log(M)log(M/δ)​), which is polylogarithmic in the number of observables and independent of the dimension of the unknown state, thereby answering Aaronson's original question while also providing an exponential improvement in the prior best dimension independent sample complexity of shadow tomography from Sinha (STOC 2025) and, more recently, Chen, O'Donnell, Pelecanos, and Wright. Our approach first reduces the general shadow tomography problem to a finite-ensemble estimation problem via a minimax argument. We then develop an observable-independent protocol that repeatedly applies the pretty-good measurement while updating the prior distribution over the finite ensemble according to the measurement outcomes. A tail analysis of the resulting estimation error yields simultaneous accuracy guarantees for all observables and a cubic-logarithmic upper bound. We also introduce a refined recovery-label measurement for the same finite ensemble, which yields the bound in our main theorem.
Shadow tomography for quantum states provides a sample efficient approach for predicting the properties of quantum systems when the properties are restricted to expectation values of 222-outcome POVMs. However, these shadow tomography procedures yield poor bounds if there are more than 2 outcomes per measurement. In this paper, we consider a general problem of learning properties from unknown quantum states: given an unknown ddd-dimensional quantum state ρρρ and MMM unknown quantum measurements M1,...,MM\mathcal{M}_1,...,\mathcal{M}_MM1​,...,MM​ with K≥2K\geq 2K≥2 outcomes, estimating the probability distribution for applying Mi\mathcal{M}_iMi​ on ρρρ to within total variation distance εεε. Compared to the special case when K=2K=2K=2, we need to learn unknown distributions instead of values. We develop an online shadow tomography procedure that solves this problem with high success probability requiring O~(Klog⁡2Mlog⁡d/ε4)\tilde{O}(K\log^2M\log d/ε^4)O~(Klog2Mlogd/ε4) copies of ρρρ. We further prove an information-theoretic lower bound that at least Ω(min⁡{d2,K+log⁡M}/ε2)Ω(\min\{d^2,K+\log M\}/ε^2)Ω(min{d2,K+logM}/ε2) copies of ρρρ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on MMM and ddd and is sample-optimal for the dependence on KKK.
We give the first tight sample complexity bounds for shadow tomography and classical shadows in the regime where the target error is below some sufficiently small inverse polynomial in the dimension of the Hilbert space. Formally we give a protocol that, given any m∈Nm\in\mathbb{N}m∈N and ε≤O(d−12)ε\le O(d^{-12})ε≤O(d−12), measures O(log⁡(m)/ε2)O(\log(m)/ε^2)O(log(m)/ε2) copies of an unknown mixed state ρ∈Cd×dρ\in\mathbb{C}^{d\times d}ρ∈Cd×d and outputs a classical description of ρρρ which can then be used to estimate any collection of mmm observables to within additive accuracy εεε. Previously, even for the simpler task of shadow tomography -- where the mmm observables are known in advance -- the best known rates either scaled benignly but suboptimally in all of m,d,εm, d, εm,d,ε, or scaled optimally in ε,mε, mε,m but had additional polynomial factors in ddd for general observables. Intriguingly, we also show via dimensionality reduction, that we can rescale εεε and ddd to reduce to the regime where ε≤O(d−1/2)ε\le O(d^{-1/2})ε≤O(d−1/2). Our algorithm draws upon representation-theoretic tools recently developed in the context of full state tomography.
We study the sample complexity of shadow tomography in the high-precision regime under realistic measurement constraints. Given an unknown ddd-dimensional quantum state ρρρ and a known set of observables {Oi}i=1m\{O_i\}_{i=1}^m{Oi​}i=1m​, the goal is to estimate expectation values {tr(Oiρ)}i=1m\{\mathrm{tr}(O_iρ)\}_{i=1}^m{tr(Oi​ρ)}i=1m​ to accuracy εεε in LpL_pLp​-norm, using possibly adaptive measurements that act on O(polylog(d))O(\mathrm{polylog}(d))O(polylog(d)) number of copies of ρρρ at a time. We focus on the regime where εεε is below an instance-dependent threshold. Our main contribution is an instance-optimal characterization of the sample complexity as Θ~(Γp/ε2)\tildeΘ(Γ_p/ε^2)Θ~(Γp​/ε2), where ΓpΓ_pΓp​ is a function of {Oi}i=1m\{O_i\}_{i=1}^m{Oi​}i=1m​ defined via an optimization formula involving the inverse Fisher information matrix. Previously, tight bounds were known only in special cases, e.g. Pauli shadow tomography with L∞L_\inftyL∞​-norm error. Concretely, we first analyze a simpler oblivious variant where the goal is to estimate an observable of the form ∑i=1mαiOi\sum_{i=1}^m α_i O_i∑i=1m​αi​Oi​ with ∥α∥q=1\|α\|_q = 1∥α∥q​=1 (where qqq is dual to ppp) revealed after the measurement. For single-copy measurements, we obtain a sample complexity of Θ(Γpob/ε2)Θ(Γ^{\mathrm{ob}}_p/ε^2)Θ(Γpob​/ε2). We then show Θ~(Γp/ε2)\tildeΘ(Γ_p/ε^2)Θ~(Γp​/ε2) is necessary and sufficient for the original problem, with the lower bound applying to unbiased, bounded estimators. Our upper bounds rely on a two-step algorithm combining coarse tomography with local estimation. Notably, Γ∞ob=Γ∞Γ^{\mathrm{ob}}_\infty = Γ_\inftyΓ∞ob​=Γ∞​. In both cases, allowing ccc-copy measurements improves the sample complexity by at most Ω(1/c)Ω(1/c)Ω(1/c). Our results establish a quantitative correspondence between quantum learning and metrology, unifying asymptotic metrological limits with finite-sample learning guarantees.
We study the problems of quantum tomography and shadow tomography using measurements performed on individual, identical copies of an unknown d -dimensional state. We first revisit known lower bounds [ 23 ] on quantum tomography with accuracy ε in trace distance, when the measurement choices are independent of previously observed outcomes, i.e., they are nonadaptive. We give a succinct proof of these results through the χ 2 -divergence between suitable distributions. Unlike prior work, we do not require that the measurements be given by rank-one operators. This leads to stronger lower bounds when the learner uses measurements with a constant number of outcomes (e.g., two-outcome measurements). In particular, this rigorously establishes the optimality of the folklore “Pauli tomography” algorithm in terms of its sample complexity. We also derive novel bounds of Ω(r2d/ϵ2)\Omega (r^2 d/\epsilon ^2)Ω(r2d/ϵ2) and Ω(r2d2/ϵ2)\Omega (r^2 d^2/\epsilon ^2)Ω(r2d2/ϵ2) for learning rank r states using arbitrary and constant-outcome measurements, respectively, in the nonadaptive case. In addition to the sample complexity, a resource of practical significance for learning quantum states is the number of unique measurement settings required (i.e., the number of different measurements used by an algorithm, each possibly with an arbitrary number of outcomes). Motivated by this consideration, we employ concentration of measure of χ 2 -divergence of suitable distributions to extend our lower bounds to the case where the learner performs possibly adaptive measurements from a fixed set of exp⁡(O(d))\exp (O(d))exp(O(d)) possible measurements. This implies in particular that adaptivity does not give us any advantage using single-copy measurements that are efficiently implementable. We also obtain a similar bound in the case where the goal is to predict the expectation values of a given sequence of observables, a task known as shadow tomography. Finally, in the case of adaptive, single-copy measurements implementable with polynomial-size circuits, we prove that a straightforward strategy based on computing sample means of the given observables is optimal.
We give a shadow tomography algorithm that uses O(log⁡(m)log⁡(m/δ)/ε2)O(\log(m) \log(m/δ) / \varepsilon^2)O(log(m)log(m/δ)/ε2) copies of the input state. Specifically, given a quantum state ρρρ and observables O1,…,OmO_1,\ldots,O_mO1​,…,Om​, the algorithm outputs estimates of tr(Oiρ)\mathrm{tr}(O_iρ)tr(Oi​ρ) for all i∈[m]i\in[m]i∈[m] that are all accurate to within ±ε\pm\varepsilon±ε, except with probability at most δδδ. The key idea is to work in the average-case setting: for a random state drawn from a known prior, a pretty good measurement on O(log⁡(m)/ε2)O(\log(m)/\varepsilon^2)O(log(m)/ε2) copies gives a constant-error guarantee on all but a small fraction of the prior. Boosting this guarantee gives an average-case shadow tomography algorithm, and the minimax principle then (non-constructively) implies the existence of a measurement with the same worst-case guarantee. We also identify an obstruction to further improvement with this approach. While pretty good measurements amplify success over the randomness of the prior, they do not amplify success over the randomness of measurement. This issue appears to be inherent, and gives rise to the remaining log⁡(m)\log(m)log(m) factor separating the complexity of the quantum problem from its classical counterpart.