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arXiv:2405.13939

Principal eigenstate classical shadows

Daniel Grier, Hakop Pashayan, Luke Schaeffer

Published May 22, 2024• 3 citations
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Abstract

Given many copies of an unknown quantum state $ρ$, we consider the task of learning a classical description of its principal eigenstate. Namely, assuming that $ρ$ has an eigenstate $|ϕ\rangle$ with (unknown) eigenvalue $λ> 1/2$, the goal is to learn a (classical shadows style) classical description of $|ϕ\rangle$ which can later be used to estimate expectation values $\langle ϕ|O| ϕ\rangle$ for any $O$ in some class of observables. We consider the sample-complexity setting in which generating a copy of $ρ$ is expensive, but joint measurements on many copies of the state are possible. We present a protocol for this task scaling with the principal eigenvalue $λ$ and show that it is optimal within a space of natural approaches, e.g., applying quantum state purification followed by a single-copy classical shadows scheme. Furthermore, when $λ$ is sufficiently close to $1$, the performance of our algorithm is optimal--matching the sample complexity for pure state classical shadows.

DOI
10.48550/arxiv.2405.13939

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Given many copies of an unknown quantum state ρρρ, we consider the task of learning a classical description of its principal eigenstate. Namely, assuming that ρρρ has an eigenstate ∣ϕ⟩|ϕ\rangle∣ϕ⟩ with (unknown) eigenvalue λ>1/2λ> 1/2λ>1/2, the goal is to learn a (classical shadows style) classical description of ∣ϕ⟩|ϕ\rangle∣ϕ⟩ which can later be used to estimate expectation values ⟨ϕ∣O∣ϕ⟩\langle ϕ|O| ϕ\rangle⟨ϕ∣O∣ϕ⟩ for any OOO in some class of observables. We consider the sample-complexity setting in which generating a copy of ρρρ is expensive, but joint measurements on many copies of the state are possible. We present a protocol for this task scaling with the principal eigenvalue λλλ and show that it is optimal within a space of natural approaches, e.g., applying quantum state purification followed by a single-copy classical shadows scheme. Furthermore, when λλλ is sufficiently close to 111, the performance of our algorithm is optimal--matching the sample complexity for pure state classical shadows.
We consider the classical shadows task for pure states in the setting of both joint and independent measurements. The task is to measure few copies of an unknown pure state ρ in order to learn a classical description which suffices to later estimate expectation values of observables. Specifically, the goal is to approximate Tr(Oρ) for any Hermitian observable O to within additive error ϵ provided Tr(O2)≤B and ‖O‖=1 . Our main result applies to the joint measurement setting, where we show Θ~(Bϵ−1+ϵ−2) samples of ρ are necessary and sufficient to succeed with high probability. The upper bound is a quadratic improvement on the previous best sample complexity known for this problem. For the lower bound, we see that the bottleneck is not how fast we can learn the state but rather how much any classical description of ρ can be compressed for observable estimation. In the independent measurement setting, we show that O(Bdϵ−1+ϵ−2) samples suffice. Notably, this implies that the random Clifford measurements algorithm of Huang, Kueng, and Preskill, which is sample-optimal for mixed states, is not optimal for pure states. Interestingly, our result also uses the same random Clifford measurements but employs a different estimator.
We study the sample complexity of the classical shadows task: what is the fewest number of copies of an unknown state you need to measure to predict expected values with respect to some class of observables? Large joint measurements are likely required in order to minimize sample complexity, but previous joint measurement protocols only work when the unknown state is pure. We present the first joint measurement protocol for classical shadows whose sample complexity scales with the rank of the unknown state. In particular we prove O(rB/ε2)\mathcal O(\sqrt{rB}/ε^2)O(rB​/ε2) samples suffice, where rrr is the rank of the state, BBB is a bound on the squared Frobenius norm of the observables, and εεε is the target accuracy. In the low-rank regime, this is a nearly quadratic advantage over traditional approaches that use single-copy measurements. We present several intermediate results that may be of independent interest: a solution to a new formulation of classical shadows that captures functions of non-identical input states; a generalization of a ``nice'' Schur basis used for optimal qubit purification and quantum majority vote; and a measurement strategy that allows us to use local symmetries in the Schur basis to avoid intractable Weingarten calculations in the analysis.
Efficiently learning expectation values of unknown quantum states via classical shadows has become an important primitive in both theoretical and experimental aspects of quantum computation. Typically, classical shadow protocols involve randomised measurements induced by sampling uniformly randomly from a compact group, a situation which is now quite well understood. In this work we go beyond this standard assumption, studying the classical shadow protocols occasioned by sampling uniformly randomly from the so-called compact symmetric spaces. We uncover a unifying theory of such protocols, extending the extent to which the general theory of classical shadows is understood at a mathematical level. Interestingly, for the estimation of observables sampled from certain distributions we further find that some of these protocols allow for slight improvements in sample-complexity over existing shadow schemes. As part of our technical analysis we obtain ε\varepsilonε-approximate relative-error kkk-designs over the nnn-qubit symplectic group in depth O(nk+log⁡(1/ε))\mathcal{O}(nk+\log(1/\varepsilon))O(nk+log(1/ε)), which is optimal with respect to nnn, and may be of some independent interest.
Classical shadow estimation (CSE) is a powerful tool for learning the properties of quantum states and quantum processes. Here we consider the CSE task for quantum unitary channels. By querying an unknown unitary channel U multiple times in quantum experiments, the goal is to learn a classical description from which one can accurately predict many different linear properties of the channel, i.e., the expectation values of arbitrary observables measured on the output of U upon arbitrary input states. Based on collective measurements on multiple systems, we propose a query-efficient protocol for this task, whose query complexity has a quadratic advantage over the previous best approach for this problem, and almost saturates the information-theoretic lower bound. To further enhance practicality, we also present a variant protocol using only single-copy measurements, which still offers much better query performance than previous protocols that do not use quantum memory, and can serve as a key subroutine for learning an arbitrary unknown Hamiltonian from dynamics. In addition to linear properties of unitary channels, our protocol can also be applied to simultaneously predict many nonlinear properties, such as out-of-time-ordered correlators.
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 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.