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arXiv:2410.04373

Computational Complexity of Learning Efficiently Generatable Pure States

Taiga Hiroka, Min-Hsiu Hsieh

Published Oct 6, 2024• 0 citations
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Abstract

Understanding the computational complexity of learning efficient classical programs in various learning models has been a fundamental and important question in classical computational learning theory. In this work, we study the computational complexity of quantum state learning, which can be seen as a quantum generalization of distributional learning introduced by Kearns et.al [STOC94]. Previous works by Chung and Lin [TQC21], and Bădescu and O$'$Donnell [STOC21] study the sample complexity of the quantum state learning and show that polynomial copies are sufficient if unknown quantum states are promised efficiently generatable. However, their algorithms are inefficient, and the computational complexity of this learning problem remains unresolved. In this work, we study the computational complexity of quantum state learning when the states are promised to be efficiently generatable. We show that if unknown quantum states are promised to be pure states and efficiently generateable, then there exists a quantum polynomial time algorithm $A$ and a language $L \in PP$ such that $A^L$ can learn its classical description. We also observe the connection between the hardness of learning quantum states and quantum cryptography. We show that the existence of one-way state generators with pure state outputs is equivalent to the average-case hardness of learning pure states. Additionally, we show that the existence of EFI implies the average-case hardness of learning mixed states.

DOI
10.48550/arxiv.2410.04373

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Understanding the computational complexity of learning efficient classical programs in various learning models has been a fundamental and important question in classical computational learning theory. In this work, we study the computational complexity of quantum state learning, which can be seen as a quantum generalization of distributional learning introduced by Kearns et.al [STOC94]. Previous works by Chung and Lin [TQC21], and Bădescu and O′'′Donnell [STOC21] study the sample complexity of the quantum state learning and show that polynomial copies are sufficient if unknown quantum states are promised efficiently generatable. However, their algorithms are inefficient, and the computational complexity of this learning problem remains unresolved. In this work, we study the computational complexity of quantum state learning when the states are promised to be efficiently generatable. We show that if unknown quantum states are promised to be pure states and efficiently generateable, then there exists a quantum polynomial time algorithm AAA and a language L∈PPL \in PPL∈PP such that ALA^LAL can learn its classical description. We also observe the connection between the hardness of learning quantum states and quantum cryptography. We show that the existence of one-way state generators with pure state outputs is equivalent to the average-case hardness of learning pure states. Additionally, we show that the existence of EFI implies the average-case hardness of learning mixed states.
The relationship between cryptography and learning theory has long been a central theme in the foundations of theoretical computer science: cryptographic primitives can imply hardness of learning, while hardness of learning can in turn be used to construct cryptographic schemes. Recent works have begun exploring analogous connections in the quantum setting, relating the average-case hardness of learning quantum states (AHL) to cryptographic primitives such as one-way state generators (OWSG). Despite recent progress exploring this for pure states, the relationship for mixed states has remained an open question. In this work, we prove that the existence of AHL for mixed quantum states is equivalent to the existence of inefficiently verifiable one-way state generators (IV-OWSGs). As a consequence, this relates mixed-state AHL to EFI pairs. Moreover, as a corollary of existing results, we obtain a separation between IV-OWSGs and OWSGs relative to the SWAP oracle.
We generalize the PAC (probably approximately correct) learning model to the quantum world by generalizing the concepts from classical functions to quantum processes, defining the problem of \emph{PAC learning quantum process}, and study its sample complexity. In the problem of PAC learning quantum process, we want to learn an ϵ\epsilonϵ-approximate of an unknown quantum process c∗c^*c∗ from a known finite concept class CCC with probability 1−δ1-\delta1−δ using samples {(x1,c∗(x1)),(x2,c∗(x2)),… }\{(x_1,c^*(x_1)),(x_2,c^*(x_2)),\dots\}{(x1​,c∗(x1​)),(x2​,c∗(x2​)),…}, where {x1,x2,… }\{x_1,x_2, \dots\}{x1​,x2​,…} are computational basis states sampled from an unknown distribution DDD and {c∗(x1),c∗(x2),… }\{c^*(x_1),c^*(x_2),\dots\}{c∗(x1​),c∗(x2​),…} are the (possibly mixed) quantum states outputted by c∗c^*c∗. The special case of PAC-learning quantum process under constant input reduces to a natural problem which we named as approximate state discrimination, where we are given copies of an unknown quantum state c∗c^*c∗ from an known finite set CCC, and we want to learn with probability 1−δ1-\delta1−δ an ϵ\epsilonϵ-approximate of c∗c^*c∗ with as few copies of c∗c^*c∗ as possible. We show that the problem of PAC learning quantum process can be solved with O(log⁡∣C∣+log⁡(1/δ)ϵ2)O\left(\frac{\log|C| + \log(1/ \delta)} { \epsilon^2}\right)O(ϵ2log∣C∣+log(1/δ)​) samples when the outputs are pure states and O(log⁡3∣C∣(log⁡∣C∣+log⁡(1/δ))ϵ2)O\left(\frac{\log^3 |C|(\log |C|+\log(1/ \delta))} { \epsilon^2}\right)O(ϵ2log3∣C∣(log∣C∣+log(1/δ))​) samples if the outputs can be mixed. Some implications of our results are that we can PAC-learn a polynomial sized quantum circuit in polynomial samples and approximate state discrimination can be solved in polynomial samples even when concept class size ∣C∣|C|∣C∣ is exponential in the number of qubits, an exponentially improvement over a full state tomography.
Morohoshi, Nakayama, Manabe, and Mitarai proposed a physically motivated quantum machine learning problem in which the goal is to predict quantities of the form Tr⁡[f(H)ρ]\operatorname{Tr}[f(H)ρ]Tr[f(H)ρ] from classical descriptions of a Hamiltonian HHH and a quantum state ρρρ, where fff is an unknown function. We call this problem Hamiltonian function learning in this paper. They constructed an efficient quantum learning algorithm under suitable conditions, while leaving a rigorous proof of average-case classical hardness open. In this paper, we rigorously prove the average-case classical hardness for two distribution-specific Hamiltonian function learning problems for fcos⁡,π(λ)=cos⁡(πλ)f_{\cos,π}(λ)=\cos(πλ)fcos,π​(λ)=cos(πλ) and fexp⁡,β(λ)=e−βλf_{\exp,β}(λ)=e^{-βλ}fexp,β​(λ)=e−βλ discussed in the paper of Morohoshi et al. under the assumption of the average-case hardness of factoring random RSA moduli. More specifically, we show that an efficient classical randomized learner under squared loss whose output hypotheses are evaluable in classical polynomial time for either problem would yield a classical randomized polynomial-time algorithm for factoring random RSA moduli.
Learning quantum states from measurement data is a central problem in quantum information and computational complexity. In this work, we study the problem of learning to generate mixed states on a finite-dimensional lattice. Motivated by recent developments in mixed state phases of matter, we focus on arbitrary states in the trivial phase. A state belongs to the trivial phase if there exists a shallow preparation channel circuit under which local reversibility is preserved throughout the preparation. We prove that any mixed state in this class can be efficiently learned from measurement access alone. Specifically, given copies of an unknown trivial phase mixed state, our algorithm outputs a shallow local channel circuit that approximately generates this state in trace distance. The sample complexity and runtime are polynomial (or quasi-polynomial) in the number of qubits, assuming constant (or polylogarithmic) circuit depth and gate locality. Importantly, the learner is not given the original preparation circuit and relies only on its existence. Our results provide a structural foundation for quantum generative models based on shallow channel circuits. In the classical limit, our framework also inspires an efficient algorithm for classical diffusion models using only a polynomial overhead of training and generation.
There is a large body of work studying what forms of computational hardness are needed to realize classical cryptography. In particular, one-way functions and pseudorandom generators can be built from each other, and thus require equivalent computational assumptions to be realized. Furthermore, the existence of either of these primitives implies that P ≠ N P , which gives a lower bound on the necessary hardness.One can also define versions of each of these primitives with quantum output: respectively one-way state generators and pseudorandom state generators. Unlike in the classical setting, it is not known whether either primitive can be built from the other. Although it has been shown that pseudorandom state generators for certain parameter regimes can be used to build one-way state generators, the implication has not been previously known in full generality. Furthermore, to the best of our knowledge, the existence of one-way state generators has no known implications in complexity theory.We show that pseudorandom states compressing n bits to log ⁡ n + 1 qubits can be used to build one-way state generators and pseudorandom states compressing n bits to ω ( log ⁡ n ) qubits are one-way state generators. This is a nearly optimal result since pseudorandom states with fewer than c log ⁡ n -qubit output can be shown to exist unconditionally. We also show that any one-way state generator can be broken by a quantum algorithm with classical access to a P P oracle.An interesting implication of our results is that a t ( n ) -copy one-way state generator exists unconditionally, for every t ( n ) = o ( n / log ⁡ n ) . This contrasts nicely with the previously known fact that O ( n ) -copy one-way state generators require computational hardness. We also outline a new route towards a black-box separation between one-way state generators and quantum bit commitments.
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