We describe algorithms to obtain an approximate classical description of a d-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for ℓq-norm error up to logarithmic factors. As a special case, we show that it takes applications of the unitaries to obtain an ε-ℓ2-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. We characterize the query complexity for obtaining Schatten q-norm estimates of a rank-r mixed state, up to polylogarithmic factors. In particular, we show that a trace-norm (q = 1) estimate can be obtained with queries. This improves (assuming our stronger input model) the ε-dependence over the works of O'Donnell and Wright (STOC 2016) and Haah et al. (IEEE Trans. Inf. Theory, 63.9, 2017), that use a joint measurement on copies of the state. To our knowledge, the most sample-efficient results for pure-state tomography come from setting the rank to 1 in generic mixed-state tomography algorithms, which can require a large amount of computing resources. We describe sample-optimal algorithms for pure states that are simple and fast to implement. Along the way we show that an ℓ∞-norm estimate of a normalized vector induces a (slightly worse) ℓq-norm estimate for that vector, without losing a dimension-dependent factor in the precision. We also develop an unbiased and symmetric version of phase estimation, where the probability distribution of the estimate is centered around the true value. Finally, we give an efficient method for estimating multiple expectation values, improving over the recent result by Huggins et al. (arXiv:2111.09283) when the measurement operators do not fully overlap. More specifically, we show that for E1,…, Em normalized measurement operators, all expectation values Tr(Ejρ) can be efficiently learned up to error ε with applications of a state-preparation unitary for a purification of ρ. * The full version of the paper can be accessed at https://arxiv.org/abs/2207.08800
We describe algorithms to obtain an approximate classical description of a d-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for ℓq-norm error up to logarithmic factors. As a special case, we show that it takes applications of the unitaries to obtain an ε-ℓ2-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. We characterize the query complexity for obtaining Schatten q-norm estimates of a rank-r mixed state, up to polylogarithmic factors. In particular, we show that a trace-norm (q = 1) estimate can be obtained with queries. This improves (assuming our stronger input model) the ε-dependence over the works of O'Donnell and Wright (STOC 2016) and Haah et al. (IEEE Trans. Inf. Theory, 63.9, 2017), that use a joint measurement on copies of the state. To our knowledge, the most sample-efficient results for pure-state tomography come from setting the rank to 1 in generic mixed-state tomography algorithms, which can require a large amount of computing resources. We describe sample-optimal algorithms for pure states that are simple and fast to implement. Along the way we show that an ℓ∞-norm estimate of a normalized vector induces a (slightly worse) ℓq-norm estimate for that vector, without losing a dimension-dependent factor in the precision. We also develop an unbiased and symmetric version of phase estimation, where the probability distribution of the estimate is centered around the true value. Finally, we give an efficient method for estimating multiple expectation values, improving over the recent result by Huggins et al. (arXiv:2111.09283) when the measurement operators do not fully overlap. More specifically, we show that for E1,…, Em normalized measurement operators, all expectation values Tr(Ejρ) can be efficiently learned up to error ε with applications of a state-preparation unitary for a purification of ρ. * The full version of the paper can be accessed at https://arxiv.org/abs/2207.08800
A longstanding belief in quantum tomography is that estimating a mixed state is far harder than estimating a pure state. This is borne out in the mathematics, where mixed state algorithms have always required more sophisticated techniques to design and analyze than pure state algorithms. We present a new approach to tomography demonstrating that, contrary to this belief, state-of-the-art mixed state tomography follows easily and naturally from pure state algorithms.
We analyze the following strategy: given n copies of an unknown state ρ, convert them into copies of a purification ∣ρ⟩; run a pure state tomography algorithm to produce an estimate of ∣ρ⟩; and output the resulting estimate of ρ. The purification subroutine was recently discovered via the "acorn trick" of Tang, Wright, and Zhandry. With this strategy, we obtain the first tomography algorithm which is sample-optimal in all parameters. For a rank-rd-dimensional state, it uses n=O((rd+log(1/δ))/ε) samples to output an estimate which is ε-close in fidelity with probability at least 1−δ. This algorithm also uses poly(n) gates, making it the first gate-efficient tomography algorithm which is sample-optimal even in terms of the dimension d alone. Moreover, with this method we recover essentially all results on mixed state tomography, including its applications to tomography with limited entanglement, classical shadows, and quantum metrology. Our proofs are simple, closing the gap in conceptual difficulty between mixed and pure tomography. Our results also clarify the role of entangled measurement in mixed state tomography: the only step of the algorithm which requires entanglement across copies is the purification step, suggesting that, for tomography, the reason entanglement is useful is for consistent purification.
We describe algorithms to obtain an approximate classical description of a d-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for ℓq-norm error up to logarithmic factors. As a special case, we show that it takes Θ(d/ε) applications of the unitaries to obtain an ε-ℓ2-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. In this model we give an efficient algorithm for obtaining Schatten q-norm estimates of a rank-r mixed state, giving query upper bounds that are close to optimal. In particular, we show that a trace-norm (q=1) estimate can be obtained with O(dr/ε) queries. This improves (assuming our stronger input model) the ε-dependence over the algorithm of Haah et al.\ (2017) that uses a joint measurement on O(dr/ε2) copies of the state. To our knowledge, the most sample-efficient results for pure-state tomography come from setting the rank to 1 in generic mixed-state tomography algorithms, which can be computationally demanding. We describe sample-optimal algorithms for pure states that are easy and fast to implement. Along the way we show that an ℓ∞-norm estimate of a normalized vector induces a (slightly worse) ℓq-norm estimate for that vector, without losing a dimension-dependent factor in the precision. We also develop an unbiased and symmetric version of phase estimation, where the probability distribution of the estimate is centered around the true value. Finally, we give an efficient method for estimating multiple expectation values, improving over the recent result by Huggins et al.\ (2021) when the measurement operators do not fully overlap.
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 E1,…,EM, the goal is to estimate all expectation values {Tr(ρEi)}i=1M to additive accuracy ε 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(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.
We study the estimation of an unknown quantum channel E with input dimension d1, output dimension d2 and Kraus rank at most r. We establish a connection between the query complexities in two models: (i) access to E, and (ii) access to a random dilation of E. Specifically, we show that for parallel (possibly coherent) testers, access to dilations does not help. This is proved by constructing a local tester that uses n queries to E yet faithfully simulates the tester with n queries to a random dilation. As application, we show that:
- O(rd1d2/ε2) queries to E suffice for channel tomography to within diamond norm error ε.
Moreover, when rd2=d1, we show that the Heisenberg scaling O(1/ε) can be achieved, even if E is not a unitary channel:
- O(min{d12.5/ε,d12/ε2}) queries to E suffice for channel tomography to within diamond norm error ε, and O(d12/ε) queries suffice for the case of Choi state trace norm error ε.
- O(min{d11.5/ε,d1/ε2}) queries to E suffice for tomography of the mixed state E(∣0⟩⟨0∣) to within trace norm error ε.
We study the problem of tomography for k-sparse quantum states. In contrast to classical distribution learning, where tight sample and time complexity bounds in terms of support size are well understood, no non-trivial bounds were previously shown for this problem. We give the first near optimal algorithm for learning n-qubit k-sparse pure quantum states, obtaining fidelity at least 1−ε with high probability using O~(k/ε) copies of the state and O~(kn/ε) time. Both bounds are optimal up to polylogarithmic factors. As an implication, we also obtain an algorithm with near optimal O~(kr/ε) sample complexity for learning k-sparse rank-r mixed states, via the random purification channel technique. Obtaining time complexity nearly matching the sample complexity, for r>1, remains an important open question.
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.