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arXiv:2404.19105 · Annual Symposium on Foundations of Computer Science

Optimal Tradeoffs for Estimating Pauli Observables

Sitan Chen, Weiyuan Gong, Qi Ye

Published Oct 27, 2024• 16 citations
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Abstract

We revisit the problem of Pauli shadow tomography: given copies of an unknown n-qubit quantum state$\rho$, estimate Tr$(P\rho)$for some set of Pauli operators$F$to within additive error$\epsilon$. This has been a popular testbed for exploring the advantage of protocols with quantum memory over those without: with enough memory to measure two copies at a time, one can use Bell sampling to estimate$\vert \text{Tr}(P\rho)$for all$P$using$O(n/\epsilon^{4})$copies, but with$k\leq n$qubits of memory,$\Omega(2^{(n-k)/3})$copies are needed. These results leave open several natural questions. How does this picture change in the physically relevant setting where one only needs to estimate a certain subset of Paulis? What is the optimal dependence on$\epsilon ?$What is the optimal tradeoff between quantum memory and sample complexity? We answer all of these questions: •For any subset$A$of Paulis and any family of measurement strategies, we completely characterize the optimal sample complexity, up to$\log\vert A\vert$factors. •We show any protocol that makes poly$(n)$-copy measure-ments must make$\Omega(1/\epsilon^{4})$measurements. •For any protocol that makes poly$(n)$-copy measurements and only has$k < n$qubits of memory, we show that$\tilde{\Theta}(\min\{2^{n}/\epsilon^{2},2^{n-k}/\epsilon^{4}\})$copies are necessary and sufficient. The protocols we propose can also estimate the actual values$\text{Tr}(P\rho)$, rather than just their absolute values as in prior work. Additionally, as a byproduct of our techniques, we establish tight bounds for the task of purity testing and show that it exhibits an intriguing phase transition not present in the memory-sample tradeoff for Pauli shadow tomography.

DOI
10.1109/focs61266.2024.00072

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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.
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 addresses the task of efficiently predicting many expectation values of an unknown quantum state from randomized measurements on comparatively few copies. Existing analyses promise large scaling advantages, but the optimal strategies realizing these guarantees are not always known, and the required measurements are potentially challenging to implement on current hardware. We address this gap for Pauli observables by computing optimal sample-complexity parameters and constructing optimal measurement strategies under realistic resource constraints. We focus on memoryless protocols, where each copy is measured only once, and on measurements with bounded interaction range. Our approach reduces the problem to the analysis of graph parameters of the frustration graph encoding the Pauli anticommutation relations. We provide efficient numerical methods for the general case and analytically prove that Clifford measurements are optimal in many situations. This includes all perfect graphs, all single-qubit, all two-qubit measurement scenarios, and more. Applied to Hamiltonian energy estimation, our framework yields constructive strategies and improved variance bounds for molecular benchmarks.
We study the power of quantum memory for learning properties of quantum systems and dynamics, which is of great importance in physics and chemistry. Many state-of-the-art learning algorithms require access to an additional external quantum memory. While such a quantum memory is not required a priori, in many cases, algorithms that do not utilize quantum memory require much more data than those which do. We show that this trade-off is inherent in a wide range of learning problems. Our results include the following: •We show that to perform shadow tomography on annnn-qubit stateρ\rhoρwithMMMobservables, any algorithm without quantum memory requiresΩ~(min⁡(M,2n))\tilde{\Omega}(\min(M, 2^{n}))Ω~(min(M,2n))samples ofρ\rhoρin the worst case. Up to log factors, this matches the upper bound of [1], and completely resolves an open question in [2], [3]. •We establish exponential separations between algorithms with and without quantum memory for purity testing, distinguishing scrambling and depolarizing evolutions, and uncovering symmetry in physical dynamics. Our separations improve and generalize prior work of [4] by allowing for a broader class of algorithms without quantum memory. •We give the first tradeoff between quantum memory and sample complexity. More precisely, we prove that to estimate absolute values of allnnn-qubit Pauli observables, algorithms withk<nk < nk<nqubits of quantum memory require at leastΩ(2(n−k)/3)\Omega(2^{(n-k)/3})Ω(2(n−k)/3)samples, but there is an algorithm usingnnn-qubit quantum memory which only requiresO(n)\mathcal{O}(n)O(n)samples. The separations we show are sufficiently large and could already be evident, for instance, with tens of qubits. This provides a concrete path towards demonstrating real-world advantage for learning algorithms with quantum memory.
We study the power of quantum memory for learning properties of quantum systems and dynamics, which is of great importance in physics and chemistry. Many state-of-the-art learning algorithms require access to an additional external quantum memory. While such a quantum memory is not required a priori, in many cases, algorithms that do not utilize quantum memory require much more data than those which do. We show that this trade-off is inherent in a wide range of learning problems. Our results include the following: (1) We show that to perform shadow tomography on an nnn-qubit state rho with MMM observables, any algorithm without quantum memory requires Ω(min⁡(M,2n))Ω(\min(M, 2^n))Ω(min(M,2n)) samples of rho in the worst case. Up to logarithmic factors, this matches the upper bound of [HKP20] and completely resolves an open question in [Aar18, AR19]. (2) We establish exponential separations between algorithms with and without quantum memory for purity testing, distinguishing scrambling and depolarizing evolutions, as well as uncovering symmetry in physical dynamics. Our separations improve and generalize prior work of [ACQ21] by allowing for a broader class of algorithms without quantum memory. (3) We give the first tradeoff between quantum memory and sample complexity. We prove that to estimate absolute values of all nnn-qubit Pauli observables, algorithms with k<nk < nk<n qubits of quantum memory require at least Ω(2(n−k)/3)Ω(2^{(n-k)/3})Ω(2(n−k)/3) samples, but there is an algorithm using nnn-qubit quantum memory which only requires O(n)O(n)O(n) samples. The separations we show are sufficiently large and could already be evident, for instance, with tens of qubits. This provides a concrete path towards demonstrating real-world advantage for learning algorithms with quantum memory.
We investigate the problem of efficiently estimating expectation values of large sets of observables from copies of an unknown many-body quantum state. This task, known as shadow tomography, is crucial for quantum simulation and quantum chemistry, where the relevant observables often include k -body fermionic or k -local Pauli operators. A key goal is to achieve sample efficiency, computational efficiency, and few-copy measurements. We introduce the notion of triply efficient shadow tomography to formalize these requirements. Prior work has shown that single-copy measurements suffice for k -local Pauli operators with constant k . However, we prove that for k -body fermionic observables and the full set of n -qubit Pauli operators, single-copy protocols fail to achieve sample-efficient tomography. We then present new protocols based on measuring two copies of the state at a time that overcome these lower bounds, providing a triply efficient protocol.