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Annual Symposium on Foundations of Computer Science

Quantum learning algorithms imply circuit lower bounds

Srinivasan Arunachalam, Alex B. Grilo, Tom Gur, Igor C. Oliveira, Aarthi Sundaram

Published Feb 1, 2022• 6 citations
Publisher

Abstract

We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let$\mathfrak{C}$be a class of polynomial-size concepts, and suppose that$\mathfrak{C}$can be PAC-learned with membership queries under the uniform distribution with error$1/2 -\gamma$by a time$T$quantum algorithm. We prove that if$\gamma^{2}\cdot T \ll 2^{n} /n$, then$\mathsf{BQE}\not\subset \mathfrak{C}$, where$\mathsf{BQE} = \mathsf{BQTIME}[2^{O(n)}]$is an exponential-time analogue of$\mathsf{BQP}$. This result is optimal in both$\gamma$and$T$, since it is not hard to learn any class$\mathfrak{C}$of functions in (classical) time$T=2^{n}$(with no error), or in quantum time$T= \mathsf{poly}(n)$with error at most$1/2-\Omega(2^{-n/2})$via Fourier sampling. In other words, even a marginal quantum speedup over these generic learning algorithms would lead to major consequences in complexity lower bounds. As a consequence, our result shows that the study of quantum learning speedups is intimately connected to fundamental open problems about algorithms, quantum computing, and complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges, since extracting computational hardness from a quantum computation is inherently more complicated. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.

DOI
10.1109/focs52979.2021.00062

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Quantum learning algorithms imply circuit lower bounds

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We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let $\mathfrak{C}$ be a class of polynomial-size concepts, and suppose that $\mathfrak{C}$ can be PAC-learned with membership queries under the uniform distribution with error $1/2 - γ$ by a time $T$ quantum algorithm. We prove that if $γ^2 \cdot T \ll 2^n/n$, then $\mathsf{BQE} \nsubseteq \mathfrak{C}$, where $\mathsf{BQE} = \mathsf{BQTIME}[2^{O(n)}]$ is an exponential-time analogue of $\mathsf{BQP}$. This result is optimal in both $γ$ and $T$, since it is not hard to learn any class $\mathfrak{C}$ of functions in (classical) time $T = 2^n$ (with no error), or in quantum time $T = \mathsf{poly}(n)$ with error at most $1/2 - Ω(2^{-n/2})$ via Fourier sampling. In other words, even a marginal improvement on these generic learning algorithms would lead to major consequences in complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and crucially, on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.

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We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, letC\mathfrak{C}Cbe a class of polynomial-size concepts, and suppose thatC\mathfrak{C}Ccan be PAC-learned with membership queries under the uniform distribution with error1/2−γ1/2 -\gamma1/2−γby a timeTTTquantum algorithm. We prove that ifγ2⋅T≪2n/n\gamma^{2}\cdot T \ll 2^{n} /nγ2⋅T≪2n/n, thenBQE⊄C\mathsf{BQE}\not\subset \mathfrak{C}BQE⊂C, whereBQE=BQTIME[2O(n)]\mathsf{BQE} = \mathsf{BQTIME}[2^{O(n)}]BQE=BQTIME[2O(n)]is an exponential-time analogue ofBQP\mathsf{BQP}BQP. This result is optimal in bothγ\gammaγandTTT, since it is not hard to learn any classC\mathfrak{C}Cof functions in (classical) timeT=2nT=2^{n}T=2n(with no error), or in quantum timeT=poly(n)T= \mathsf{poly}(n)T=poly(n)with error at most1/2−Ω(2−n/2)1/2-\Omega(2^{-n/2})1/2−Ω(2−n/2)via Fourier sampling. In other words, even a marginal quantum speedup over these generic learning algorithms would lead to major consequences in complexity lower bounds. As a consequence, our result shows that the study of quantum learning speedups is intimately connected to fundamental open problems about algorithms, quantum computing, and complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges, since extracting computational hardness from a quantum computation is inherently more complicated. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.
We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let C\mathfrak{C}C be a class of polynomial-size concepts, and suppose that C\mathfrak{C}C can be PAC-learned with membership queries under the uniform distribution with error 1/2−γ1/2 - γ1/2−γ by a time TTT quantum algorithm. We prove that if γ2⋅T≪2n/nγ^2 \cdot T \ll 2^n/nγ2⋅T≪2n/n, then BQE⊈C\mathsf{BQE} \nsubseteq \mathfrak{C}BQE⊈C, where BQE=BQTIME[2O(n)]\mathsf{BQE} = \mathsf{BQTIME}[2^{O(n)}]BQE=BQTIME[2O(n)] is an exponential-time analogue of BQP\mathsf{BQP}BQP. This result is optimal in both γγγ and TTT, since it is not hard to learn any class C\mathfrak{C}C of functions in (classical) time T=2nT = 2^nT=2n (with no error), or in quantum time T=poly(n)T = \mathsf{poly}(n)T=poly(n) with error at most 1/2−Ω(2−n/2)1/2 - Ω(2^{-n/2})1/2−Ω(2−n/2) via Fourier sampling. In other words, even a marginal improvement on these generic learning algorithms would lead to major consequences in complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and crucially, on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.
We establish an unconditional super-polynomial separation between quantum and classical PAC learning of Boolean functions under the uniform distribution in the large error regime. For every advantage λ(n)=n−ω(1)λ(n) = n^{-ω(1)}λ(n)=n−ω(1), we construct a concept class that a polynomial-time quantum learner learns to agreement 1/2+λ(n)1/2+λ(n)1/2+λ(n) with success probability at least 1−1/nk1-1/n^k1−1/nk. The learner uses only quantum random examples and outputs a polynomial-size deterministic classical circuit. In contrast, no polynomial-time classical learner achieves agreement 1/2+λ(n)/nk1/2 + λ(n)/n^k1/2+λ(n)/nk with success probability greater than 1/nk1/n^k1/nk, even with membership queries. To achieve this, we develop a nearly-optimal quantum learner that uses quantum random examples to produce a hypothesis achieving a prescribed correlation with an arbitrary Boolean function. The learner generalizes Fourier sampling from parity hypotheses to quadratic hypotheses. While standard Fourier sampling is insensitive to the sign of the correlation, we use interference to bias sampling toward positively correlated hypotheses. This is combined with a careful analysis of the resulting distribution to boost both the learner's success probability and its agreement with the target function. We believe that the resulting quantum learning algorithm might be of independent interest. Finally, although a robust quantum PAC learning advantage holds unconditionally in the large error regime, we note that an analogous separation between quantum and classical learning in the small error regime would imply that NP⊈BPP\mathsf{NP} \nsubseteq \mathsf{BPP}NP⊈BPP.
We establish connections between state tomography, pseudorandomness, quantum state synthesis, and circuit lower bounds. In particular, let C\mathfrak{C}C be a family of non-uniform quantum circuits of polynomial size and suppose that there exists an algorithm that, given copies of ∣ψ⟩|ψ\rangle∣ψ⟩, distinguishes whether ∣ψ⟩|ψ\rangle∣ψ⟩ is produced by C\mathfrak{C}C or is Haar random, promised one of these is the case. For arbitrary fixed constant ccc, we show that if the algorithm uses at most O(2nc)O(2^{n^c})O(2nc) time and 2n0.992^{n^{0.99}}2n0.99 samples then stateBQE⊄stateC\mathsf{stateBQE} \not\subset \mathsf{state}\mathfrak{C}stateBQE⊂stateC. Here stateBQE:=stateBQTIME[2O(n)]\mathsf{stateBQE} := \mathsf{stateBQTIME}[2^{O(n)}]stateBQE:=stateBQTIME[2O(n)] and stateC\mathsf{state}\mathfrak{C}stateC are state synthesis complexity classes as introduced by Rosenthal and Yuen (ITCS 2022), which capture problems with classical inputs but quantum output. Note that efficient tomography implies a similarly efficient distinguishing algorithm against Haar random states, even for nearly exponential-time algorithms. Because every state produced by a polynomial-size circuit can be learned with 2O(n)2^{O(n)}2O(n) samples and time, or O(nω(1))O(n^{ω(1)})O(nω(1)) samples and 2O(nω(1))2^{O(n^{ω(1)})}2O(nω(1)) time, we show that even slightly non-trivial quantum state tomography algorithms would lead to new statements about quantum state synthesis. Finally, a slight modification of our proof shows that distinguishing algorithms for quantum states can imply circuit lower bounds for decision problems as well. This help sheds light on why time-efficient tomography algorithms for non-uniform quantum circuit classes has only had limited and partial progress. Our work parallels results by Arunachalam et al. (FOCS 2021) that revealed a similar connection between quantum learning of Boolean functions and circuit lower bounds for classical circuit classes, but modified for the purposes of state tomography and state synthesis.
In this paper, we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of probably approximately correct (PAC) learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results. 1. Hardness of PAC learning AC0{AC}^0AC0 and TC0{TC}^0TC0 under the uniform distribution. Our first result concerns the concept class TC0{TC}^0TC0 (resp., AC0{AC}^0AC0), the class of constant-depth, polynomial-sized circuits with unbounded fan-in majority gates (resp., AND,OR,NOT{AND}, {OR}, {NOT}AND,OR,NOT gates). We show the following: if there exists no quantum (quasi-)polynomial-time algorithm to solve the ring-learning with errors (RLWE{RLWE}RLWE) problem, then there exists no (quasi-)polynomial-time quantum learning algorithm for TC0{TC}^0TC0; and if there exists no 2O(d1/η)2^{O(d^{1/\eta})}2O(d1/η)-time quantum algorithm to solve RLWE{RLWE}RLWE with dimension d=O(polylogn)d = O(polylog n)d=O(polylogn) (for every constant η>2\eta > 2η>2), then there exists no O(nlog⁡νn)O(n^{ \log^{\nu} n} )O(nlogνn)-time quantum learning algorithm for poly(n)poly(n)poly(n)-sized AC0{AC}^0AC0 circuits (for a constant ν>0\nu>0ν>0), matching the classical upper bound of Linial, Mansour and Nisan [J. ACM, 40 (1993), pp. 607--620], where the learning algorithms are under the uniform distribution (even with access to quantum membership queries). The main technique in these results uses an explicit family of pseudorandom functions that are believed to be quantum-secure to construct concept classes that are hard to learn quantumly under the uniform distribution. 2. Hardness of learning TC20{TC}^0_2TC20​ in the PAC setting. Our second result shows that if there exists no quantum polynomial-time algorithm for the LWE{LWE}LWE problem, then there exists no polynomial-time quantum-PAC learning algorithm for the class TC20{TC}^0_2TC20​, i.e., depth-2 TC0{TC}^0TC0 circuits. The main technique in this result is to establish a connection between the quantum security of public-key encryption schemes and the learnability of a concept class that consists of decryption functions of the cryptosystem. Our results show that quantum resources do not give an exponential improvement to learning constant-depth polynomial-sized neural networks. This also gives a strong (conditional) negative answer to one of the “Ten Semi-Grand Challenges for Quantum Computing Theory" raised by Aaronson https://www.scottaaronson.com/writings/qchallenge.html, 2005.
We continue the study of the circuit class GC^0, which augments AC^0 with unbounded-fan-in gates that compute arbitrary functions inside a sufficiently small Hamming ball but must be constant outside it. While GC^0 can compute functions requiring exponential-size circuits, Kumar (CCC 2023) showed that switching-lemma lower bounds for AC^0 extend to GC^0 with no loss in parameters. We prove a parallel result for the polynomial method: any lower bound for AC^0[p] obtained via the polynomial method extends to GC^0[p] without loss in parameters. As a consequence, we show that the majority function MAJ requires depth-ddd GC^0[p] circuits of size 2Ω(n1/2(d−1))2^{Ω(n^{1/2(d-1)})}2Ω(n1/2(d−1)), matching the best-known lower bounds for AC^0[p]. This yields the most expressive class of non-monotone circuits for which exponential-size lower bounds are known for an explicit function. We also prove a similar result for the algorithmic method, showing that E^NP requires exponential-size GCC^0 circuits, extending a result of Williams (JACM 2014). Finally, leveraging our improved classical lower bounds, we establish the strongest known unconditional separations between quantum and classical circuit classes. We separate QNC^0 from GC^0 and GC^0[p] in various settings and show that BQLOGTIME is not contained in GC^0. As a consequence, we construct an oracle relative to which BQP lies outside uniform GC^0, extending the Raz-Tal oracle separation between BQP and PH (STOC 2019).
Although a concept class may be learnt more efficiently using quantum samples as compared with classical samples in certain scenarios, quantum learners are asymptotically no more efficient than classical ones in the quantum PAC and Agnostic learning models. Lower bounds on sample complexity in these models were previously established via quantum state identification and Fourier analysis. In this paper, we derive optimal lower bounds for quantum sample complexity in both models via an information-theoretic approach. The proofs are arguably simpler, and the same ideas can potentially be used to derive optimal bounds for other problems in quantum learning theory. We then turn to a quantum analogue of the Coupon Collector problem, a classic problem from probability theory also of importance in the study of PAC learning. The quantum sample complexity of this problem has been characterised up to constant factors. First, we show that the information-theoretic approach mentioned above provably does not yield the optimal lower bound. As a by-product, we get a natural ensemble of pure states in arbitrarily high dimensions which are not easily (simultaneously) distinguishable, whereas the ensemble has close to maximal Holevo information. Second, we discover that the information-theoretic approach yields an asymptotically optimal bound for an approximation variant of the problem. Finally, we derive a sharper lower bound for the Quantum Coupon Collector problem via the generalised Holevo-Curlander bounds. All the aspects of the problem we study rest on properties of the spectrum of the associated Gram matrix, which may be of independent interest.