We continue our analysis of: (i) "Quantum tomography", i.e., learning a quantum state, i.e., the quantum generalization of learning a discrete probability distribution; (ii) The distribution of Young diagrams output by the RSK algorithm on random words. Regarding (ii), we introduce two powerful new tools: first, a precise upper bound on the expected length of the longest union of k disjoint increasing subsequences in a random length-n word with letter distribution α1 ≥ α2 ≥ … ≥ αd. Our bound has the correct main term and second-order term, and holds for all n, not just in the large-n limit. Second, a new majorization property of the RSK algorithm that allows one to analyze the Young diagram formed by the lower rows λk, λk+1, … of its output. These tools allow us to prove several new theorems concerning the distribution of random Young diagrams in the nonasymptotic regime, giving concrete error bounds that are optimal, or nearly so, in all parameters. As one example, we give a fundamentally new proof of the celebrated fact that the expected length of the longest increasing sequence in a random length-n permutation is bounded by 2√n. This is the k = 1, αi ≡ 1/d, d → ∞ special case of a much more general result we prove: the expected length of the kth Young diagram row produced by an α-random word is αk n ± 2√αkd n.
We continue our analysis of: (i) "Quantum tomography", i.e., learning a quantum state, i.e., the quantum generalization of learning a discrete probability distribution; (ii) The distribution of Young diagrams output by the RSK algorithm on random words. Regarding (ii), we introduce two powerful new tools: first, a precise upper bound on the expected length of the longest union of k disjoint increasing subsequences in a random length-n word with letter distribution α1 ≥ α2 ≥ … ≥ αd. Our bound has the correct main term and second-order term, and holds for all n, not just in the large-n limit. Second, a new majorization property of the RSK algorithm that allows one to analyze the Young diagram formed by the lower rows λk, λk+1, … of its output. These tools allow us to prove several new theorems concerning the distribution of random Young diagrams in the nonasymptotic regime, giving concrete error bounds that are optimal, or nearly so, in all parameters. As one example, we give a fundamentally new proof of the celebrated fact that the expected length of the longest increasing sequence in a random length-n permutation is bounded by 2√n. This is the k = 1, αi ≡ 1/d, d → ∞ special case of a much more general result we prove: the expected length of the kth Young diagram row produced by an α-random word is αk n ± 2√αkd n.
In this survey, we give an introduction to the problem of learning and testing quantum states from multiple copies.Special attention is paid to the tasks of estimating the eigenvalues of an unknown state, testing whether a state is maximally mixed, and performing full quantum state tomography.We focus on the representation theory-based approached, developed in [ARS88, KW01, HM02, Har05, Key06, CM06, Chr06, CHW07, MW16, OW21, OW16, HHJ + 17, OW17, BOW19, CLM + 21, AISW19] and elsewhere.Examining this approach also leads to interesting new developments regarding the probabilistic combinatorics of longest increasing subsequences.Note: Modulo minor revisions, Sections 1 to 12 were written jointly with John Wright (Berkeley), and a somewhat condensed version of these sections appeared under the title A primer on the statistics of longest increasing subsequences and quantum states in the Sept. 2017 edition of SIGACT News.Contents 1. Spanish cryptograms 44 2. Quantum contraptions 46 3. Longest increasing subsequences: Robinson, Schensted, Knuth 48 4. Symmetric properties of probabilities: classical vs. quantum 51 5.The RSK bijection and Schur symmetric polynomials 53 6.Two majorization theorems for the RSK algorithm 56 7. Probabilistic combinatorics of longest increasing subsequences 58 8. Mechanics of quantum mechanics 61 9. Noncommutative probability 65 10.Testing for the uniform distribution/maximally mixed state 67 11.Representation theory gives a nice basis for observables 71 12. Symmetric group characters on small conjugacy classes 75
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) and 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)) 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.
In quantum purity amplification, one is given n copies of a noisy quantum state ρ∈Cd×d and asked to prepare k copies of its principal eigenstate ∣vd⟩. Several prior works have derived information-theoretically optimal algorithms for this problem, but the bounds they prove are only shown in the asymptotic regime as the number of samples n tends to infinity. In this paper, we establish the following nonasymptotic guarantee: if ρ's eigenvalues are sorted p1≤⋯≤pd and pd−1<pd, then \begin{equation*}
n = O\Big(k + \frac{k}δ \cdot \frac{1-p_d}{(p_d-p_{d-1})^2}\Big)
\end{equation*} copies suffice to output a state with fidelity at least 1−δ with ∣vd⊗k⟩. Our bound holds for arbitrary spectra, and is independent of the dimension d. In the case of depolarizing noise, our finite-sample guarantee matches the optimal asymptotic scaling. Our proof is based on the combinatorics of random Young diagrams.
How many samples of a quantum state are required to learn a complete description of it? As we will see in this thesis, the fine-grained answer depends on the measurements available to the learner, but in general it is at least Ω(d^2/ϵ^2) where d is the dimension of the state and ϵ the trace distance accuracy. Optimal algorithms for this task -- known as quantum state tomography -- make use of powerful, yet highly impractical entangled measurements, where some joint measurement is performed on all copies of the state. What can be accomplished without such measurements, where one must perform measurements on individual copies of the states? \n \nIn Chapter 2 we show a relationship between the recently proposed quantum online learning framework and quantum state tomography. Specifically, we show that tomography can be accomplished using online learning algorithms in a black-box manner and O(d^4/ϵ^4) two-outcome measurements on separate copies of the state. The interpretation of this approach is that the experimentalist uses informative measurements to teach the learner by helping it make "mistakes" on measurements as early as possible. \n \nWe move on to proving lower bounds on tomography in Chapter 3. First, we review a known lower bound for entangled measurements as well as a Ω(d^3/ϵ^2) lower bound in the setting where non-entangled measurements are made non-adaptively, both due to Ref.[18]. We then derive a novel bound of Ω(d^4/ϵ^2) samples when the learner is further restricted to observing a constant number of outcomes (e.g., two-outcome measurements). This implies that the folklore "Pauli tomography" algorithm is optimal in this setting. \n \nUnderstanding the power of adaptive measurements, where measurement choices can depend on previous outcomes, is currently an open problem. In Chapter 4 we present two scenarios in which adapting on previous outcomes makes no difference to the number of samples required. In the first, the learner is limited to adapting on at most o(d^2/ϵ^2) of the previous outcomes. In the second, measurements are drawn from some set of at most exp(O(d)) measurements. In particular, this second lower bound implies that adaptivity makes no difference in the regime of efficiently implementable measurements, in the context of quantum computing. \n \nFinally, we apply the above technique to the problems of classical shadows and shadow tomography to obtain similar lower bounds. Here, one is interested only in determining the expectations of some fixed set of observables. We once again find that, for the worst-case input of observables, adaptivity makes no difference to the sample complexity when considering efficient, non-entangled measurements. As a corollary, we find a straightforward algorithm for shadow tomography is optimal in this setting.
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 2-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 d-dimensional quantum state ρ and M unknown quantum measurements M1,...,MM with K≥2 outcomes, estimating the probability distribution for applying Mi on ρ to within total variation distance ε. Compared to the special case when K=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~(Klog2Mlogd/ε4) copies of ρ. We further prove an information-theoretic lower bound that at least Ω(min{d2,K+logM}/ε2) copies of ρ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on M and d and is sample-optimal for the dependence on K.
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.