Private learning implies quantum stability

1 Introduction . Learning an unknown quantum state ρ , given copies of the state is a fundamental task in quantum computing , often referred to as tomography . Tomography is of great practical interest since it helps in tasks such as verifying entanglement , understanding correlations , and is useful for calibrating , understanding and controlling noise in quantum devices . In the last few years , questions about the fundamental limits of this task have gained a lot of theoretical attention : how many copies of an n-qubit quantum state ρ are required to estimate the density matrix of ρ up to small error ? Recent ∗https : //stanford.edu/∼yquek 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . breakthrough results of [ 1 , 2 , 3 ] showed that Θ ( 22n/ε2 ) copies of ρ are necessary and sufficient to learn ρ up to trace distance ε . Unfortunately , this exponential scaling in complexity hampers practical applications of tomography ; the best known experimental implementation of full-state quantum tomography has been for a 10-qubit quantum state [ 4 ] . Simultaneously , the use of machine learning for quantum states has gained lot of traction in quantum computing . In this area , the goal is to use heuristics to learn an unknown quantum state , often with the motivation of physical implementation . Provable results in this area are very few , but interest is widespread due to the number of emerging quantum hardware devices . Given the fundamental importance of tomography and the natural connection to learning , Aaronson [ 5 ] formally defined a variant of tomography : PAC learning quantum states . This problem brings together two ripe exciting topics : classical machine learning and quantum computing . Following his work , there have been a few other papers that have introduced models of learning quantum states with various constraints . In this work , our main contribution is to show implications between several seemingly different quantum learning models . The main technicality is extending known results in classical learning theory from Boolean functions to real-valued functions with noisy labels – a setting motivated by learning quantum states , but with potential extensions to learning other physical systems ( e.g . [ 6 ] ) . Along the way we also show how certain implications that hold for classical Boolean functions do not hold true in the real-valued setting ( which could be of independent classical interest ) . See Figure 1 for a summary of our results . 1.1 Background : Models of interest . To explain our main results , we start by introducing some learning models of interest . We state them in the language of quantum state learning . However , these are at their core , classical learning models for real-valued functions with imprecise feedback , and the translation is stated precisely in Section 2.2 . We also formally translate them to the classical learning setting in Supplementary material ( Section A ) , which we omit here due to space constraints . PAC learning . Valiant ’ s Probably Approximately Correct ( PAC ) learning , lays the foundation for computational learning theory . Aaronson [ 5 ] considered learning quantum states in the PAC model . In this model , let ρ ∈ C be an unknown quantum state ( picked from a known concept class C of states ) and let D : E → [ 0 , 1 ] be an arbitrary unknown distribution over all possible 2-outcome measurements E. Suppose a quantum learner obtains training examples ( Ei , Tr ( ρEi ) ) where Ei is drawn from D , and the goal is to output σ such that with probability ≥ 0.99 , σ satisfies PrE∼D [ |Tr ( σE ) −Tr ( ρE ) | ≤ ζ ] ≥ 1−α . How many training examples suffice for such a ( ζ , α ) -PAC learner ? In answer , he showed that the number of examples necessary and sufficient to learn C is captured by the fat-shattering dimension of C. PAC learning with Differential privacy . A well-studied area of computer science is differential privacy ( DP ) ( which says that an algorithm should behave “ approximately '' the same given two datasets that differ in one element ) . This notion can be extended to the quantum realm , where we ask that the quantum PAC learner proposed above is also differentially private , wherein given two datasets S = { ( Ei , Tr ( ρEi ) ) } i , S′ = { ( E′i , Tr ( ρE′i ) ) } i such that there exists a unique i such that Ei 6= E′i , then a quantum ( γ , δ ) -DP PAC learning algorithm needs to satisfy Pr [ A ( S ) = σ ] ≤ eγ Pr [ A ( S′ ) = σ ] + δ , where A ( S ) is the output of A on input S. 2 Communication complexity . Consider the standard one-way communication model between Alice and Bob . Suppose Alice has a quantum state ρ ( unknown to Bob ) and Bob has an unknown ( to Alice ) measurement E. The goal of Bob is to output an approximation of Tr ( ρE ) if only Alice is allowed to communicate to Bob . A trivial strategy for this communication task is for Alice to send a classical description of ρ , but can we do better ? If so , how many bits of communication suffice for this task ? Online learning . Several features of the PAC learning model and tomography are somewhat artificial : first , the assumption that the measurements ( training examples ) are drawn from the same unknown distribution D that the learner will be evaluated on , which does not account for adversarial or changing environments , and secondly , it may be infeasible to possess T -fold tensor copies of the unknown quantum state ρ , rather we may only be able to obtain sequential copies of it . The quantum 2Our notion of DP differs from the notion of DP proposed by [ 7 ] . They consider DP measurements with respect to a class of product states , whereas here we require DP with respect to the dataset { ( Ei , Tr ( Eiρ ) } i. online learning model addresses these aspects . Online learning consists of repeating the following rounds of interaction : a learner obtains a copy of ρ , maintains a local σ which is its guess of ρ , obtains a description of measurement operator Ei ( possibly adversarially ) and predicts the value of yi = Tr ( ρEi ) . Subsequently it receives as feedback an ε-approximation of yi . On every round , if the learner ’ s prediction satisfies |Tr ( σEi ) − yi| ≤ ε then it is correct , otherwise it has made a mistake . The goal of the learner is the following : minimize m so that after making m mistakes ( not necessarily consecutively ) , it makes a correct prediction on all future rounds . Importantly , while the goal in the above model is to make real-valued predictions yi , it departs from the real-valued online learning literature in allowing for ε-imprecision in the feedback . This imprecision is inherent to all learning settings where the feedback is generated by a statistical algorithm ( in our setting , the feedback arises from processing the outcomes of quantum measurements ) , and this generalization has non-trivial implications , as we show . Working in this model , Aaronson [ 8 ] showed that for learning the class of all quantum states , it suffices to let m be at most sequential fat-shattering dimension of C ( a combinatorial parameter which was originally introduced by Rakhlin et al . [ 9 ] ) . Motivating question . All these learning models can be seen as variants of full-state tomography , and are known to require exponentially fewer resources than tomography . A natural question is : Is there a relation between these learning models , communication and combinatorial parameters ? . Understanding this question classically in the context of Boolean functions has received tremendous attention in computational learning theory and theoretical computer science in the last two years . There have been several papers establishing various connections [ 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 ] , just to cite a few . However , these papers leave two important questions open : 1 . Do these results apply to learning quantum states ? 2 . Do the classical results ( for Boolean functions ) also hold for real-valued functions ? Ours is the first work that studies the first question . In the process , we not only explore how implications between Boolean learning models translate over to the quantum setting , but we also introduce new notions of ( classically ) privately learning quantum states and quantum stability ( linked to privacy ) , contributing another perspective to an ongoing discussion of what differentially-private quantum computation could look like [ 19 , 20 , 7 ] . The second question indirectly received attention recently in NeurIPS ’ 20 by Jung et al . [ 11 ] for multi-class learning , but their work left several open ends ( when translated to real-valued learning ) which we tie up here ( see Section 3.4 for a more in-depth comparison ) . 2 Overview of main results . 2.1 Quantum results . To condense our ( affirmative ) answer to the first question above , we derive a series of implications going through all these models , starting from differentially-private PAC learning to online learning to quantum stability ( our conceptual contribution which we define and discuss below ) . Taking a step back , quantum online learning and DP PAC quantum learning seem very different on the surface . Online learning ensures that eventually ( after m mistakes ) we learn the state approximately , but makes no guarantees on when the last of m mistakes occurs with respect to the series of examples seen . DP PAC learning is not online – it separates the learning into train ( offline ) and test ( online ) phases , and also introduces a distribution , D , from which measurements are drawn . Ultimately DP PAC learning says that after seeing T measurements from D , we have ( privately ) learned the state . We show that in fact DP PAC learning sample complexity can be lower bounded by the sequential fat-shattering dimension which also characterizes the complexity of online learning [ 9 ] . We give a summary of our results in Figure 1 . Also , we say an algorithm is pure DP ( resp . approximate DP ) when δ = 0 in our definition ( resp . δ > 0 ) . We remark that only a few arrows are efficient in both sample and time complexity , otherwise these implications are primarily information-theoretic . Conceptual contribution . The main center piece in establishing these connections is the concept of quantum stability , which is the new conceptual contribution in this work . Intuitively , we say a quantum learning algorithm is stable if , for an unknown state ρ , given a set of noisy labelled examples from a distribution D , there exists one state σ such that , with “ high '' probability , the output of the learning algorithm is “ close '' to σ . More formally , we say a quantum learning algorithm A is ( T , ε , η ) -stable with respect to distribution D over a set of orthogonal 2-outcome measurements if , given T many labelled examples S consisting of Ei drawn from D and ζ-approximations of Tr ( ρEi ) , there exists a state σ such that Pr [ A ( S ) ∈ BM ( ε , σ ) ] ≥ η , ( 1 ) where the probability is taken over the examples in S and BM ( ε , σ ) is the ball of states ε-close to σ with respect toM , i.e. , BM ( ε , σ ) = { σ′ : |Tr ( Eσ ) − Tr ( Eσ′ ) | < ε for every E ∈ M } . In other words , quantum stability means that up to an ε-distance , there is some hypothesis state σ that is output by A with “ high '' ( at least η ) probability . A classical analog of stability can be written as : a classical learning algorithm for concept class C : X → [ 0 , 1 ] ( run on examples { ( xi , ĉ ( xi ) ) } where |ĉ ( xi ) − c ( xi ) | < ζ , c ∈ C ) is stable if there exists a hypothesis f such that Pr [ A ( S ) ∈ T ( ε , f ) ] ≥ η , where T ( ε , f ) : = { g ∈ C : |g ( x ) − f ( x ) | < ε for every x ∈ X } . Consistent stability implies learnability . While we will make this precise later , the significance of an algorithm A being stable is that σ , the output state at the ‘ center of the ball ’ , is a good hypothesis for estimating measurement probabilities ( and hence A is a good learner ) . This is not at all obvious from the definition of stability , which does not inherently require that this σ is a good approximation of ρ . Yet , it turns out that if A is a stable and consistent learner ( i.e. , its output does not contradict any of the training examples it has seen ) , σ has low loss with respect to D. This means that , using hypothesis σ to predict outcomes of future measurements drawn from distribution D as Tr ( Eσ ) will yield ε-accurate predictions with high probability .

Summary: This paper presents a number of connections between private learning, online learning and stability in the quantum and real-valued settings, generalizing recent work of Bun, Livni and Moran '20 in the Boolean-valued setting. The main results are:  - An algorithm (termed RSOA) that accomplishes online learning of real-valued functions (requiring feedback only within an $\epsilon$ precision) with a mistake bound given by the sequential fat-shattering (sfat) dimension of the class  - A notion of real-valued and quantum stability (generalizing Bun et al's "global stability"), along with a proof that online learnability implies this kind of stability  - A no-go result for (one approach for) converting stability to approximate PAC learnability  - An improved shadow tomography result with sample complexity in terms of the sfat dimension  - Bounds on the sfat dimension of quantum state classes in terms of Holevo information  Many of the key notions and proofs are natural generalizations of the Boolean and (exact) real-valued analogs as studied e.g. in Bun et al '20 and Rakhlin et al '10. The concrete applications to quantum state learning are novel and interesting, and address some questions raised in earlier works such as Aaronson et al '18.