Post-Quantum Zero Knowledge, Revisited (or: How to Do Quantum Rewinding Undetectably)

Lombardi, Alex; Ma, Fermi; Spooner, Nicholas

doi:10.48550/arxiv.2111.12257

Cited by 3 publications

(6 citation statements)

References 28 publications

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“…(3) Finally, QSP/QSVT have recently been applied to a wide variety of subfields, both in and out of quantum information. These include Hamiltonian simulation [3], phase estimation [2], quantum zero-knowledge proofs [10], classical quantum-inspired machine learning algorithms [9], semi-definite programming [25], quantum adiabatic methods [26], the approximation of correlation functions [27], the approximation of fidelity [6], recovery maps [7], and fast inversion of linear systems [28]. Efforts continue to bring further computational problems into the fold of QSP/QSVT.…”

Section: A Prior Workmentioning

confidence: 99%

“…This includes Hamiltonian simulation [3,5], search [1], phase estimation [2], quantum walks [1], fidelity estimation [6], sophisticated techniques for measurement [7], and channel discrimination [8]. QSVT has found purchase in surprisingly disparate subfields, from undergirding a general theory of quantuminspired classical algorithms for low-rank machine learning [9], to quantum cryptographic protocols with zero-knowledge properties [10].…”

Section: Introductionmentioning

confidence: 99%

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Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Rossi,

Chuang

2022

Preprint

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Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a single oracle, and say nothing about computing joint properties of two or more oracles; these can be far cheaper to compute given the ability to pit oracles against one another coherently.This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desired stable multivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected.M-QSP is numerically stable and inherits the approximative efficiency of its single-variable analogue; however, its underlying theory is substantively more complex. M-QSP provides one bridge from quantum algorithms to a rich theory of algebraic geometry over multiple variables. The unique ability of M-QSP to obliviously approximate joint functions of multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms.

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Section: A Prior Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Rossi,

Chuang

2022

Preprint

View full text Add to dashboard Cite

show abstract

Section: Prior Workmentioning

confidence: 99%

“…This includes Hamiltonian simulation [5,3], search [1], phase estimation [2], quantum walks [1], fidelity estimation [6], sophisticated techniques for measurement [7], and channel discrimination [8]. QSVT has found purchase in surprisingly disparate subfields, from undergirding a general theory of quantum-inspired classical algorithms for low-rank machine learning [9], to quantum cryptographic protocols with zero-knowledge properties [10].…”

Section: Introductionmentioning

confidence: 99%

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Rossi

Chuang

2022

Quantum

View full text Add to dashboard Cite

Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a single oracle, saying nothing about computing joint properties of two or more oracles; these can be far cheaper to determine given an ability to pit oracles against one another coherently. This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desired stable multivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons, and remain numerically stable and efficient. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected. The unique ability of M-QSP to obliviously approximate joint functions of multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms, and provides a bridge from quantum algorithms to algebraic geometry.

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“…In [Wat09], the reason for this is the fact that the space complexity of the simulator depends on the communication complexity of the protocol which in turn is some function of the security parameter. In [CCY21,CCLY21b,LMS21], the simulator runs the verifier coherently multiple times and thus, the space complexity additionally depends on the number of iterations.…”

Section: Introductionmentioning

confidence: 99%

Post-Quantum Zero-Knowledge with Space-Bounded Simulation

Ananth,

Grilo

2022

Preprint

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The traditional definition of quantum zero-knowledge stipulates that the knowledge gained by any quantum polynomial-time verifier in an interactive protocol can be simulated by a quantum polynomial-time algorithm. One drawback of this definition is that it allows the simulator to consume significantly more computational resources than the verifier. We argue that this drawback renders the existing notion of quantum zero-knowledge not viable for certain settings, especially when dealing with near-term quantum devices.In this work, we initiate a fine-grained notion of post-quantum zero-knowledge that is more compatible with near-term quantum devices. We introduce the notion of (𝑠, 𝑓 ) space-bounded quantum zero-knowledge. In this new notion, we require that an 𝑠-qubit malicious verifier can be simulated by a quantum polynomial-time algorithm that uses at most 𝑓 (𝑠)-qubits, for some function 𝑓 (•), and no restriction on the amount of the classical memory consumed by either the verifier or the simulator.We explore this notion and establish both positive and negative results:• For verifiers with logarithmic quantum space 𝑠 and (arbitrary) polynomial classical space, we show that (𝑠, 𝑓 )-space-bounded QZK, for 𝑓 (𝑠) = 2𝑠, can be achieved based on the existence of post-quantum one-way functions. Moreover, our protocol runs in constant rounds.• For verifiers with super-logarithmic quantum space 𝑠, assuming the existence of postquantum secure one-way functions, we show that (𝑠, 𝑓 )-space-bounded QZK protocols, with fully black box simulation (classical analogue of black-box simulation) can only be achieved for languages in BQP.

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Post-Quantum Zero Knowledge, Revisited (or: How to Do Quantum Rewinding Undetectably)

Cited by 3 publications

References 28 publications

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Post-Quantum Zero-Knowledge with Space-Bounded Simulation

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