arXiv subjects

Henry Yuen

Publications and source records attributed to Henry Yuen.

At least 37 records · Page 2Linked to original sources

arXiv2024

An efficient quantum parallel repetition theorem and applications

We prove a tight parallel repetition theorem for $3$-message computationally-secure quantum interactive protocols between an efficient challenger and an efficient adversary. We also prove under plausible assumptions that the security of $4$-message computationally secure protocols does not generally decrease under parallel repetition. These mirror the classical results of Bellare, Impagliazzo, and Naor [BIN97]. Finally, we prove that all quantum argument systems can be generically compiled to an equivalent $3$-message argument system, mirroring the transformation for quantum proof systems [KW00, KKMV07]. As immediate applications, we show how to derive hardness amplification theorems for quantum bit commitment schemes (answering a question of Yan [Yan22]), EFI pairs (answering a question of Brakerski, Canetti, and Qian [BCQ23]), public-key quantum money schemes (answering a question of Aaronson and Christiano [AC13]), and quantum zero-knowledge argument systems. We also derive an XOR lemma [Yao82] for quantum predicates as a corollary.

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arXiv2024

Pseudorandom unitaries with non-adaptive security

Pseudorandom unitaries (PRUs) are ensembles of efficiently implementable unitary operators that cannot be distinguished from Haar random unitaries by any quantum polynomial-time algorithm with query access to the unitary. We present a simple PRU construction that is a concatenation of a random Clifford unitary, a pseudorandom binary phase operator, and a pseudorandom permutation operator. We prove that this PRU construction is secure against non-adaptive distinguishers assuming the existence of quantum-secure one-way functions. This means that no efficient quantum query algorithm that is allowed a single application of $U^{\otimes \mathrm{poly}(n)}$ can distinguish whether an $n$-qubit unitary $U$ was drawn from the Haar measure or our PRU ensemble. We conjecture that our PRU construction remains secure against adaptive distinguishers, i.e. secure against distinguishers that can query the unitary polynomially many times in sequence, not just in parallel.

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arXiv2023

Testing and Learning Quantum Juntas Nearly Optimally

We consider the problem of testing and learning quantum $k$-juntas: $n$-qubit unitary matrices which act non-trivially on just $k$ of the $n$ qubits and as the identity on the rest. As our main algorithmic results, we give (a) a $\widetilde{O}(\sqrt{k})$-query quantum algorithm that can distinguish quantum $k$-juntas from unitary matrices that are "far" from every quantum $k$-junta; and (b) a $O(4^k)$-query algorithm to learn quantum $k$-juntas. We complement our upper bounds for testing quantum $k$-juntas and learning quantum $k$-juntas with near-matching lower bounds of $Ω(\sqrt{k})$ and $Ω(\frac{4^k}{k})$, respectively. Our techniques are Fourier-analytic and make use of a notion of influence of qubits on unitaries.

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arXiv2023

Pseudorandom Strings from Pseudorandom Quantum States

We study the relationship between notions of pseudorandomness in the quantum and classical worlds. Pseudorandom quantum state generator (PRSG), a pseudorandomness notion in the quantum world, is an efficient circuit that produces states that are computationally indistinguishable from Haar random states. PRSGs have found applications in quantum gravity, quantum machine learning, quantum complexity theory, and quantum cryptography. Pseudorandom generators, on the other hand, a pseudorandomness notion in the classical world, is ubiquitous to theoretical computer science. While some separation results were known between PRSGs, for some parameter regimes, and PRGs, their relationship has not been completely understood. In this work, we show that a natural variant of pseudorandom generators called quantum pseudorandom generators (QPRGs) can be based on the existence of logarithmic output length PRSGs. Our result along with the previous separations gives a better picture regarding the relationship between the two notions. We also study the relationship between other notions, namely, pseudorandom function-like state generators and pseudorandom functions. We provide evidence that QPRGs can be as useful as PRGs by providing cryptographic applications of QPRGs such as commitments and encryption schemes. Our primary technical contribution is a method for pseudodeterministically extracting uniformly random strings from Haar-random states.

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arXiv2023

Rigidity of superdense coding

The famous superdense coding protocol of Bennett and Wiesner demonstrates that it is possible to communicate two bits of classical information by sending only one qubit and using a shared EPR pair. Our first result is that an arbitrary protocol for achieving this task (where there are no assumptions on the sender's encoding operations or the dimension of the shared entangled state) is locally equivalent to the canonical Bennett-Wiesner protocol. In other words, the superdense coding task is rigid. In particular, we show that the sender and receiver only use additional entanglement (beyond the EPR pair) as a source of classical randomness. We also investigate several questions about higher-dimensional superdense coding, where the goal is to communicate one of $d^2$ possible messages by sending a $d$-dimensional quantum state, for general dimensions $d$. Unlike the $d=2$ case (i.e. sending a single qubit), there can be inequivalent superdense coding protocols for higher $d$. We present concrete constructions of inequivalent protocols, based on constructions of inequivalent orthogonal unitary bases for all $d > 2$. Finally, we analyze the performance of superdense coding protocols where the encoding operators are independently sampled from the Haar measure on the unitary group. Our analysis involves bounding the distinguishability of random maximally entangled states, which may be of independent interest.

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arXiv2023

Pseudorandom (Function-Like) Quantum State Generators: New Definitions and Applications

Pseudorandom quantum states (PRS) are efficiently constructible states that are computationally indistinguishable from being Haar-random, and have recently found cryptographic applications. We explore new definitions, new properties and applications of pseudorandom states, and present the following contributions: 1. New Definitions: We study variants of pseudorandom function-like state (PRFS) generators, introduced by Ananth, Qian, and Yuen (CRYPTO'22), where the pseudorandomness property holds even when the generator can be queried adaptively or in superposition. We show feasibility of these variants assuming the existence of post-quantum one-way functions. 2. Classical Communication: We show that PRS generators with logarithmic output length imply commitment and encryption schemes with classical communication. Previous constructions of such schemes from PRS generators required quantum communication. 3. Simplified Proof: We give a simpler proof of the Brakerski--Shmueli (TCC'19) result that polynomially-many copies of uniform superposition states with random binary phases are indistinguishable from Haar-random states. 4. Necessity of Computational Assumptions: We also show that a secure PRS with output length logarithmic, or larger, in the key length necessarily requires computational assumptions.

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arXiv2023

stateQIP = statePSPACE

Complexity theory traditionally studies the hardness of solving classical computational problems. In the quantum setting, it is also natural to consider a different notion of complexity, namely the complexity of physically preparing a certain quantum state. We study the relation between two such state complexity classes: statePSPACE, which contains states that can be generated by space-uniform polynomial-space quantum circuits, and stateQIP, which contains states that a polynomial-time quantum verifier can generate by interacting with an all-powerful untrusted quantum prover. The latter class was recently introduced by Rosenthal and Yuen (ITCS 2022), who proved that statePSPACE $\subseteq$ stateQIP. Our main result is the reverse inclusion, stateQIP $\subseteq$ statePSPACE, thereby establishing equality of the two classes and providing a natural state-complexity analogue to the celebrated QIP = PSPACE theorem of Jain, et al. (J. ACM 2011). To prove this, we develop a polynomial-space quantum algorithm for solving a large class of exponentially large "PSPACE-computable" semidefinite programs (SDPs), which also prepares an optimiser encoded in a quantum state. Our SDP solver relies on recent block-encoding techniques from quantum algorithms, demonstrating that these techniques are also useful for complexity theory. Using similar techniques, we also show that optimal prover strategies for general quantum interactive protocols can be implemented in quantum polynomial space. We prove this by studying an algorithmic version of Uhlmann's theorem and establishing an upper bound on the complexity of implementing Uhlmann transformations.

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arXiv2023

On the (Im)plausibility of Public-Key Quantum Money from Collision-Resistant Hash Functions

Public-key quantum money is a cryptographic proposal for using highly entangled quantum states as currency that is publicly verifiable yet resistant to counterfeiting due to the laws of physics. Despite significant interest, constructing provably-secure public-key quantum money schemes based on standard cryptographic assumptions has remained an elusive goal. Even proposing plausibly-secure candidate schemes has been a challenge. These difficulties call for a deeper and systematic study of the structure of public-key quantum money schemes and the assumptions they can be based on. Motivated by this, we present the first black-box separation of quantum money and cryptographic primitives. Specifically, we show that collision-resistant hash functions cannot be used as a black-box to construct public-key quantum money schemes where the banknote verification makes classical queries to the hash function. Our result involves a novel combination of state synthesis techniques from quantum complexity theory and simulation techniques, including Zhandry's compressed oracle technique.

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arXiv2022

An Improved Sample Complexity Lower Bound for (Fidelity) Quantum State Tomography

We show that $Ω(rd/ε)$ copies of an unknown rank-$r$, dimension-$d$ quantum mixed state are necessary in order to learn a classical description with $1 - ε$ fidelity. This improves upon the tomography lower bounds obtained by Haah, et al. and Wright (when closeness is measured with respect to the fidelity function).

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arXiv2022

Unitary property testing lower bounds by polynomials

We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods. Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between $\mathsf{QMA}$ and $\mathsf{QMA(2)}$, a long standing question in quantum complexity theory.

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arXiv2022

Quantum soundness of testing tensor codes

A locally testable code is an error-correcting code that admits very efficient probabilistic tests of membership. Tensor codes provide a simple family of combinatorial constructions of locally testable codes that generalize the family of Reed-Muller codes. The natural test for tensor codes, the axis-parallel line vs. point test, plays an essential role in constructions of probabilistically checkable proofs. We analyze the axis-parallel line vs. point test as a two-prover game and show that the test is sound against quantum provers sharing entanglement. Our result implies the quantum-soundness of the low individual degree test, which is an essential component of the MIP* = RE theorem. Our proof also generalizes to the infinite-dimensional commuting-operator model of quantum provers.

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arXiv2022

MIP*=RE

We show that the class MIP* of languages that can be decided by a classical verifier interacting with multiple all-powerful quantum provers sharing entanglement is equal to the class RE of recursively enumerable languages. Our proof builds upon the quantum low-degree test of (Natarajan and Vidick, FOCS 2018) and the classical low-individual degree test of (Ji, et al., 2020) by integrating recent developments from (Natarajan and Wright, FOCS 2019) and combining them with the recursive compression framework of (Fitzsimons et al., STOC 2019). An immediate byproduct of our result is that there is an efficient reduction from the Halting Problem to the problem of deciding whether a two-player nonlocal game has entangled value $1$ or at most $1/2$. Using a known connection, undecidability of the entangled value implies a negative answer to Tsirelson's problem: we show, by providing an explicit example, that the closure $C_{qa}$ of the set of quantum tensor product correlations is strictly included in the set $C_{qc}$ of quantum commuting correlations. Following work of (Fritz, Rev. Math. Phys. 2012) and (Junge et al., J. Math. Phys. 2011) our results provide a refutation of Connes' embedding conjecture from the theory of von Neumann algebras.

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arXiv2022

Quantum search-to-decision reductions and the state synthesis problem

It is a useful fact in classical computer science that many search problems are reducible to decision problems; this has led to decision problems being regarded as the $\textit{de facto}$ computational task to study in complexity theory. In this work, we explore search-to-decision reductions for quantum search problems, wherein a quantum algorithm makes queries to a classical decision oracle to output a desired quantum state. In particular, we focus on search-to-decision reductions for $\mathsf{QMA}$, and show that there exists a quantum polynomial-time algorithm that can generate a witness for a $\mathsf{QMA}$ problem up to inverse polynomial precision by making one query to a $\mathsf{PP}$ decision oracle. We complement this result by showing that $\mathsf{QMA}$-search does $\textit{not}$ reduce to $\mathsf{QMA}$-decision in polynomial-time, relative to a quantum oracle. We also explore the more general $\textit{state synthesis problem}$, in which the goal is to efficiently synthesize a target state by making queries to a classical oracle encoding the state. We prove that there exists a classical oracle with which any quantum state can be synthesized to inverse polynomial precision using only one oracle query and to inverse exponential precision using two oracle queries. This answers an open question of Aaronson from 2016, who presented a state synthesis algorithm that makes $O(n)$ queries to a classical oracle to prepare an $n$-qubit state, and asked if the query complexity could be made sublinear.

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arXiv2022

Cryptography from Pseudorandom Quantum States

Pseudorandom states, introduced by Ji, Liu and Song (Crypto'18), are efficiently-computable quantum states that are computationally indistinguishable from Haar-random states. One-way functions imply the existence of pseudorandom states, but Kretschmer (TQC'20) recently constructed an oracle relative to which there are no one-way functions but pseudorandom states still exist. Motivated by this, we study the intriguing possibility of basing interesting cryptographic tasks on pseudorandom states. We construct, assuming the existence of pseudorandom state generators that map a $λ$-bit seed to a $ω(\logλ)$-qubit state, (a) statistically binding and computationally hiding commitments and (b) pseudo one-time encryption schemes. A consequence of (a) is that pseudorandom states are sufficient to construct maliciously secure multiparty computation protocols in the dishonest majority setting. Our constructions are derived via a new notion called pseudorandom function-like states (PRFS), a generalization of pseudorandom states that parallels the classical notion of pseudorandom functions. Beyond the above two applications, we believe our notion can effectively replace pseudorandom functions in many other cryptographic applications.

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arXiv2021

Interactive Proofs for Synthesizing Quantum States and Unitaries

Whereas quantum complexity theory has traditionally been concerned with problems arising from classical complexity theory (such as computing boolean functions), it also makes sense to study the complexity of inherently quantum operations such as constructing quantum states or performing unitary transformations. With this motivation, we define models of interactive proofs for synthesizing quantum states and unitaries, where a polynomial-time quantum verifier interacts with an untrusted quantum prover, and a verifier who accepts also outputs an approximation of the target state (for the state synthesis problem) or the result of the target unitary applied to the input state (for the unitary synthesis problem); furthermore there should exist an "honest" prover which the verifier accepts with probability 1. Our main result is a "state synthesis" analogue of the inclusion $\mathsf{PSPACE} \subseteq \mathsf{IP}$: any sequence of states computable by a polynomial-space quantum algorithm (which may run for exponential time) admits an interactive protocol of the form described above. Leveraging this state synthesis protocol, we also give a unitary synthesis protocol for polynomial space-computable unitaries that act nontrivially on only a polynomial-dimensional subspace. We obtain analogous results in the setting with multiple entangled provers as well.

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arXiv2021

Nonlocal Games, Compression Theorems, and the Arithmetical Hierarchy

We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy, a classification of languages according to the complexity of arithmetical formulas defining them. It was recently shown by Ji, Natarajan, Vidick, Wright and Yuen that deciding whether the (finite-dimensional) quantum value of a nonlocal game is $1$ or at most $\frac{1}{2}$ is complete for the class $Σ_1$ (i.e., $\mathsf{RE}$). A result of Slofstra implies that deciding whether the commuting operator value of a nonlocal game is equal to $1$ is complete for the class $Π_1$ (i.e., $\mathsf{coRE}$). We prove that deciding whether the quantum value of a two-player nonlocal game is exactly equal to $1$ is complete for $Π_2$; this class is in the second level of the arithmetical hierarchy and corresponds to formulas of the form "$\forall x \, \exists y \, ϕ(x,y)$". This shows that exactly computing the quantum value is strictly harder than approximating it, and also strictly harder than computing the commuting operator value (either exactly or approximately). We explain how results about the complexity of nonlocal games all follow in a unified manner from a technique known as compression. At the core of our $Π_2$-completeness result is a new "gapless" compression theorem that holds for both quantum and commuting operator strategies. Our compression theorem yields as a byproduct an alternative proof of Slofstra's result that the set of quantum correlations is not closed. We also show how a "gap-preserving" compression theorem for commuting operator strategies would imply that approximating the commuting operator value is complete for $Π_1$.

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arXiv2021

Anchored parallel repetition for nonlocal games

We introduce a simple transformation on two-player nonlocal games, called "anchoring", and prove an exponential-decay parallel repetition theorem for all anchored games in the setting of quantum entangled players. This transformation is inspired in part by the Feige-Kilian transformation (SICOMP 2000), and has the property that if the quantum value of the original game $G$ is $v$ then the quantum value of the anchored game $G_\bot$ is $1 - (1 - α)^2 \cdot (1 - v)$ where $α$ is a parameter of the transformation. In particular the anchored game has quantum value $1$ if and only if the original game $G$ has quantum value $1$. This provides the first gap amplification technique for general two-player nonlocal games that achieves exponential decay of the quantum value.

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arXiv2020

Quantum statistical query learning

We propose a learning model called the quantum statistical learning QSQ model, which extends the SQ learning model introduced by Kearns to the quantum setting. Our model can be also seen as a restriction of the quantum PAC learning model: here, the learner does not have direct access to quantum examples, but can only obtain estimates of measurement statistics on them. Theoretically, this model provides a simple yet expressive setting to explore the power of quantum examples in machine learning. From a practical perspective, since simpler operations are required, learning algorithms in the QSQ model are more feasible for implementation on near-term quantum devices. We prove a number of results about the QSQ learning model. We first show that parity functions, (log n)-juntas and polynomial-sized DNF formulas are efficiently learnable in the QSQ model, in contrast to the classical setting where these problems are provably hard. This implies that many of the advantages of quantum PAC learning can be realized even in the more restricted quantum SQ learning model. It is well-known that weak statistical query dimension, denoted by WSQDIM(C), characterizes the complexity of learning a concept class C in the classical SQ model. We show that log(WSQDIM(C)) is a lower bound on the complexity of QSQ learning, and furthermore it is tight for certain concept classes C. Additionally, we show that this quantity provides strong lower bounds for the small-bias quantum communication model under product distributions. Finally, we introduce the notion of private quantum PAC learning, in which a quantum PAC learner is required to be differentially private. We show that learnability in the QSQ model implies learnability in the quantum private PAC model. Additionally, we show that in the private PAC learning setting, the classical and quantum sample complexities are equal, up to constant factors.

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