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.
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.
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 this work, we consider the fundamental task of quantum state certification: given copies of an unknown quantum state ρ, test whether it matches some target state σ or is ε-far from it. For certifying d-dimensional states, Θ(d/ε2) copies of ρ are known to be necessary and sufficient. However, the algorithm achieving this complexity makes fully entangled measurements over all O(d/ε2) copies of ρ. Often, one is interested in certifying states to a high precision; this makes such joint measurements intractable even for low-dimensional states. Thus, we study whether one can obtain optimal rates for quantum state certification and related testing problems while only performing measurements on t copies at once, for some 1<t≪d/ε2. While it is well-understood how to use intermediate entanglement to achieve optimal quantum state learning, the only protocol known to achieve optimal testing is the one using fully entangled measurements.
Our main result is a smooth copy complexity upper bound for state certification as a function of t, which achieves a near-optimal rate at t=d2. In the high-precision regime, i.e., for ε<d1, this is a strict improvement over the entanglement used by the aforementioned optimal protocol. We also extend our techniques to develop new algorithms for the related tasks of mixedness testing and purity estimation, and show tradeoffs achieving the optimal rates for these problems at t=d2 as well. Our algorithms are based on novel reductions from testing to learning and leverage recent advances in quantum state tomography in a non-black-box fashion. We complement our testing upper bounds with a (nearly) matching lower bound that holds in the high-precision regime, implying that one must take t≥Ω(d2) to achieve the optimal rate for certification.
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.
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 ε, estimating an unknown state on Cd of rank at most r to trace norm error ε with constant success probability requires, and is achievable with, Θ(ε2drmax{1,tr}) 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 t. Further, measuring order r2 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.
Quantum state tomography is a fundamental problem in quantum computing. Given n copies of an unknown N -qubit state ρ∈ℂ d × d , d =2 N , the goal is to learn the state up to an accuracy ε in trace distance, say with at least constant probability 0.99. We are interested in the copy complexity, the minimum number of copies of ρ needed to fulfill the task. As current quantum devices are physically limited, Pauli measurements have attracted significant attention due to their ease of implementation. However, a large gap exists in the literature for tomography with Pauli measurements. The best-known upper bound is O ( N · 12 N /ε 2 ), and no non-trivial lower bound is known besides the general single-copy lower bound of Ω(8 N /ε 2 ), achieved by hard-to-implement structured POVMs such as MUB, SIC-POVM, and uniform POVM. We have made significant progress on this long-standing problem. We first prove a stronger upper bound of O (10 N /ε 2 ). To complement it, we also obtain a lower bound of Ω(9.118 N /ε 2 ), which holds even with adaptivity. To our knowledge, this demonstrates the first known separation between Pauli measurements and structured POVMs. The new lower bound is a consequence of a novel framework for adaptive quantum state tomography with measurement constraints. The main advantage is that we can use measurement-dependent hard instances to prove tight lower bounds for Pauli measurements, while prior lower-bound techniques for tomography only work with measurement-independent constructions. Moreover, we connect the copy complexity lower bound of tomography to the eigenvalues of the measurement information channel, which governs the measurement’s capacity to distinguish between states. To demonstrate the generality of the new framework, we obtain tight bounds for adaptive quantum state tomography with k -outcome measurements, where we recover existing results and establish new ones.
Quantum process tomography, the task of learning an unknown quantum channel from black-box access, is a central problem in quantum information. In this setting, protocols with quantum memory can coherently store and jointly process quantum information obtained from multiple channel uses, whereas protocols without quantum memory must measure after each use and retain only a classical transcript of the measurement outcomes. A fundamental open question is whether quantum memory provides a query-complexity advantage even when protocols without quantum memory may adapt their experiments based on all previous outcomes with unbounded classical computational power. In this work, we show that it does. We determine the optimal query complexity of quantum process tomography without quantum memory up to a constant factor to be Θ(din3dout3/ε2), where din and dout are the channel input and output dimensions, respectively, and ε is the target diamond-norm accuracy. More precisely, we prove that any incoherent protocol for this task, including adaptive protocols, requires Ω(din3dout3/ε2) queries, even when each channel use may be assisted by arbitrary fresh ancilla, and we present a non-adaptive, ancilla-free incoherent protocol achieving the matching upper bound O(din3dout3/ε2). Our results thereby generalize the optimal sample-complexity bounds for single-copy state tomography, recovered as the special case din=1. By contrast, coherent protocols with quantum memory achieve query complexity Θ(din2dout2/ε2). Hence, our results establish a rigorous learning separation between quantum process tomography with and without quantum memory.