Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2606.16948

The Optimal Rate Function in Covariant Quantum State Tomography

Arick Grootveld, Alexander Maloney, Jason Pollack, Peixue Wu

quant-phhep-thmath-phmath.PRSubmitted Jun 15, 2026• Announced Jun 16, 2026• 0 citations
arXivHTMLPDF

Abstract

The problem of quantum tomography is to estimate an unknown quantum state $ρ$ from a measurement of $n$ copies of $ρ$. One can ask which tomography protocol, i.e.\ which choice of multi-copy measurement, gives the best possible estimate of $ρ$. To do so, we characterize tomography protocols by their \emph{rate function}, which governs the exponential rate at which a protocol assigns probability to a particular estimate $σ$ of the true state $ρ$. This rate function is a quantum mechanical generalization of the classical relative entropy between the true state and its estimate, and depends on the choice of protocol. It is bounded by the quantum relative entropy, and we show that this bound is sharp: for any $ρ$ and $σ$ we construct a family of protocols whose rate functions converge to the quantum relative entropy $D(σ\|ρ)$. We consider the family of covariant tomography protocols; these are the basis independent state estimation schemes that assume no prior information about $ρ$ and $σ$. Keyl described a specific tomography protocol based on Schur sampling, and conjectured that among all covariant tomography protocols it has the largest possible rate function for all $σ$ and $ρ$. We prove this conjecture. The resulting rate function is an annealed version of quantum relative entropy, due to the cost of learning the eigenbasis in covariant quantum state tomography.

Related papers

Dimension-Free Polylogarithmic Quantum Shadow Tomography

Fernando Granha Jeronimo, Qizhao Huang, Lenny Liu

Shadow Tomography is a fundamental problem in quantum information theory. Given multiple copies of an unknown $d$-dimensional quantum state $ρ$ and a known collection of observables $E_1,\ldots,E_M$, the goal is to estimate all expectation values $\{\text{Tr}(ρE_i)\}_{i=1}^M$ to additive accuracy $\varepsilon$ with probability at least $1-δ$. 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 $M$, 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 \[ O\left( \frac{\log(M)\log(M/δ)}{\varepsilon^2} \right), \] 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.

quant-pharXiv:2608.06345•Aug 7, 2026•0 citations

An Optimal Tradeoff between Entanglement and Copy Complexity for State Tomography

Sitan Chen, Jerry Zheng Li, Allen Liu

There has been significant interest in understanding how practical constraints on contemporary quantum devices impact the complexity of quantum learning. For the classic question of tomography, recent work tightly characterized the copy complexity for any protocol that can only measure one copy of the unknown state at a time, showing it is polynomially worse than if one can make fully-entangled measurements. While we now have a fairly complete picture of the rates for such tasks in the near-term and fault-tolerant regimes, it remains poorly understood what the landscape in between these extremes looks like, and in particular how to gracefully scale up our protocols as we transition away from NISQ. In this work, we study tomography in the natural setting where one can make measurements of t copies at a time. For sufficiently small є, we show that for any t ≤ d2, Θ(d3/√tє2) copies are necessary and sufficient to learn an unknown d-dimensional state ρ to trace distance є. This gives a smooth and optimal interpolation between the known rates for single-copy measurements and fully-entangled measurements. To our knowledge, this is the first smooth entanglement-copy tradeoff known for any quantum learning task, and for tomography, no intermediate point on this curve was known, even at t = 2. An important obstacle is that unlike the optimal single-copy protocol, the optimal fully-entangled protocol is inherently a biased estimator. This bias precludes naive batching approaches for interpolating between the two protocols. Instead, we devise a novel two-stage procedure that uses Keyl’s algorithm to refine a crude estimate for ρ based on single-copy measurements. A key insight is to use Schur-Weyl sampling not to estimate the spectrum of ρ, but to estimate the deviation of ρ from the maximally mixed state. When ρ is far from the maximally mixed state, we devise a novel quantum splitting procedure that reduces to the case where ρ is close to maximally mixed.

arXiv:2402.16353•ACM Symposium on Theory of Computing (STOC)•Jun 10, 2024•11 citations

Minimax Quantum Tomography: Estimators and Relative Entropy Bounds

Christopher Ferrie, Robin Blume-Kohout

A minimax estimator has the minimum possible error ("risk") in the worst case. We construct the first minimax estimators for quantum state tomography with relative entropy risk. The minimax risk of nonadaptive tomography scales as O(1/sqrt[N])-in contrast to that of classical probability estimation, which is O(1/N)-where N is the number of copies of the quantum state used. We trace this deficiency to sampling mismatch: future observations that determine risk may come from a different sample space than the past data that determine the estimate. This makes minimax estimators very biased, and we propose a computationally tractable alternative with similar behavior in the worst case, but superior accuracy on most states.

arXiv:1503.03100•Physical Review Letters•Mar 4, 2016•17 citations

Optimal Tradeoffs for Estimating Pauli Observables

Sitan Chen, Weiyuan Gong, Qi Ye

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.

arXiv:2404.19105•Annual Symposium on Foundations of Computer Science•Oct 27, 2024•16 citations

Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements

Ashwin Nayak, Xingyu Zhou

We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples. For sufficiently small $\varepsilon$, estimating an unknown state on $\mathbb{C}^d$ of rank at most $r$ to trace norm error $\varepsilon$ with constant success probability requires, and is achievable with, $$Θ\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)$$ samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most $t$ samples improve the complexity of algorithms making single-sample measurements by at most a factor $\sqrt{t}$. Further, measuring order $r^2$ samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on $t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.

quant-pharXiv:2609.10514•Sep 10, 2026•0 citations

Efficient quantum tomography

Ryan O’Donnell, John Wright

In the quantum state tomography problem, one wishes to estimate an unknown d-dimensional mixed quantum state ρ, given few copies. We show that O(d/ε) copies suffice to obtain an estimate ρ that satisfies ||ρ − ρ||F2 ≤ ε (with high probability). An immediate consequence is that O((ρ) · d/ε2) ≤ O(d2/ε2) copies suffice to obtain an ε-accurate estimate in the standard trace distance. This improves on the best known prior result of O(d3/ε2) copies for full tomography, and even on the best known prior result of O(d2log(d/ε)/ε2) copies for spectrum estimation. Our result is the first to show that nontrivial tomography can be obtained using a number of copies that is just linear in the dimension.

arXiv:1508.01907•ACM Symposium on Theory of Computing (STOC)•Jun 10, 2016•181 citations
The problem of quantum tomography is to estimate an unknown quantum state ρρρ from a measurement of nnn copies of ρρρ. One can ask which tomography protocol, i.e.\ which choice of multi-copy measurement, gives the best possible estimate of ρρρ. To do so, we characterize tomography protocols by their \emph{rate function}, which governs the exponential rate at which a protocol assigns probability to a particular estimate σσσ of the true state ρρρ. This rate function is a quantum mechanical generalization of the classical relative entropy between the true state and its estimate, and depends on the choice of protocol. It is bounded by the quantum relative entropy, and we show that this bound is sharp: for any ρρρ and σσσ we construct a family of protocols whose rate functions converge to the quantum relative entropy D(σ∥ρ)D(σ\|ρ)D(σ∥ρ).
We consider the family of covariant tomography protocols; these are the basis independent state estimation schemes that assume no prior information about ρρρ and σσσ. Keyl described a specific tomography protocol based on Schur sampling, and conjectured that among all covariant tomography protocols it has the largest possible rate function for all σσσ and ρρρ. We prove this conjecture. The resulting rate function is an annealed version of quantum relative entropy, due to the cost of learning the eigenbasis in covariant quantum state tomography.
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.
There has been significant interest in understanding how practical constraints on contemporary quantum devices impact the complexity of quantum learning. For the classic question of tomography, recent work tightly characterized the copy complexity for any protocol that can only measure one copy of the unknown state at a time, showing it is polynomially worse than if one can make fully-entangled measurements. While we now have a fairly complete picture of the rates for such tasks in the near-term and fault-tolerant regimes, it remains poorly understood what the landscape in between these extremes looks like, and in particular how to gracefully scale up our protocols as we transition away from NISQ. In this work, we study tomography in the natural setting where one can make measurements of t copies at a time. For sufficiently small є, we show that for any t ≤ d2, Θ(d3/√tє2) copies are necessary and sufficient to learn an unknown d-dimensional state ρ to trace distance є. This gives a smooth and optimal interpolation between the known rates for single-copy measurements and fully-entangled measurements. To our knowledge, this is the first smooth entanglement-copy tradeoff known for any quantum learning task, and for tomography, no intermediate point on this curve was known, even at t = 2. An important obstacle is that unlike the optimal single-copy protocol, the optimal fully-entangled protocol is inherently a biased estimator. This bias precludes naive batching approaches for interpolating between the two protocols. Instead, we devise a novel two-stage procedure that uses Keyl’s algorithm to refine a crude estimate for ρ based on single-copy measurements. A key insight is to use Schur-Weyl sampling not to estimate the spectrum of ρ, but to estimate the deviation of ρ from the maximally mixed state. When ρ is far from the maximally mixed state, we devise a novel quantum splitting procedure that reduces to the case where ρ is close to maximally mixed.
A minimax estimator has the minimum possible error ("risk") in the worst case. We construct the first minimax estimators for quantum state tomography with relative entropy risk. The minimax risk of nonadaptive tomography scales as O(1/sqrt[N])-in contrast to that of classical probability estimation, which is O(1/N)-where N is the number of copies of the quantum state used. We trace this deficiency to sampling mismatch: future observations that determine risk may come from a different sample space than the past data that determine the estimate. This makes minimax estimators very biased, and we propose a computationally tractable alternative with similar behavior in the worst case, but superior accuracy on most states.
We revisit the problem of Pauli shadow tomography: given copies of an unknown n-qubit quantum stateρ\rhoρ, estimate Tr(Pρ)(P\rho)(Pρ)for some set of Pauli operatorsFFFto within additive errorϵ\epsilonϵ. 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∣Tr(Pρ)\vert \text{Tr}(P\rho)∣Tr(Pρ)for allPPPusingO(n/ϵ4)O(n/\epsilon^{4})O(n/ϵ4)copies, but withk≤nk\leq nk≤nqubits of memory,Ω(2(n−k)/3)\Omega(2^{(n-k)/3})Ω(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 onϵ?\epsilon ?ϵ?What is the optimal tradeoff between quantum memory and sample complexity? We answer all of these questions: •For any subsetAAAof Paulis and any family of measurement strategies, we completely characterize the optimal sample complexity, up tolog⁡∣A∣\log\vert A\vertlog∣A∣factors. •We show any protocol that makes poly(n)(n)(n)-copy measure-ments must makeΩ(1/ϵ4)\Omega(1/\epsilon^{4})Ω(1/ϵ4)measurements. •For any protocol that makes poly(n)(n)(n)-copy measurements and only hask<nk < nk<nqubits of memory, we show thatΘ~(min⁡{2n/ϵ2,2n−k/ϵ4})\tilde{\Theta}(\min\{2^{n}/\epsilon^{2},2^{n-k}/\epsilon^{4}\})Θ~(min{2n/ϵ2,2n−k/ϵ4})copies are necessary and sufficient. The protocols we propose can also estimate the actual valuesTr(Pρ)\text{Tr}(P\rho)Tr(Pρ), 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.
We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most ttt samples. For sufficiently small ε\varepsilonε, estimating an unknown state on Cd\mathbb{C}^dCd of rank at most rrr to trace norm error ε\varepsilonε with constant success probability requires, and is achievable with, Θ(drε2max{1,rt})Θ\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)Θ(ε2dr​max{1,t​r​}) samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most ttt samples improve the complexity of algorithms making single-sample measurements by at most a factor t\sqrt{t}t​. Further, measuring order r2r^2r2 samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on ttt samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.
In the quantum state tomography problem, one wishes to estimate an unknown d-dimensional mixed quantum state ρ, given few copies. We show that O(d/ε) copies suffice to obtain an estimate ρ that satisfies ||ρ − ρ||F2 ≤ ε (with high probability). An immediate consequence is that O((ρ) · d/ε2) ≤ O(d2/ε2) copies suffice to obtain an ε-accurate estimate in the standard trace distance. This improves on the best known prior result of O(d3/ε2) copies for full tomography, and even on the best known prior result of O(d2log(d/ε)/ε2) copies for spectrum estimation. Our result is the first to show that nontrivial tomography can be obtained using a number of copies that is just linear in the dimension.