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arXiv:2401.10200 · ACM Symposium on Theory of Computing (STOC)

Quantum State Obfuscation from Classical Oracles

James Bartusek, Zvika Brakerski, Vinod Vaikuntanathan

Published Jun 10, 2024• 19 citations
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Abstract

A major unresolved question in quantum cryptography is whether it is possible to obfuscate arbitrary quantum computation. Indeed, there is much yet to understand about the feasibility of quantum obfuscation even in the classical oracle model, where one is given for free the ability to obfuscate any classical circuit. In this work, we develop a new array of techniques that we use to construct a quantum state obfuscator, a powerful notion formalized recently by Coladangelo and Gunn (arXiv:2311.07794) in their pursuit of better software copy-protection schemes. Quantum state obfuscation refers to the task of compiling a quantum program, consisting of a quantum circuit C with a classical description and an auxiliary quantum state ψ, into a functionally-equivalent obfuscated quantum program that hides as much as possible about C and ψ. We prove the security of our obfuscator when applied to any pseudo-deterministic quantum program, i.e. one that computes a (nearly) deterministic classical input / classical output functionality. Our security proof is with respect to an efficient classical oracle, which may be heuristically instantiated using quantum-secure indistinguishability obfuscation for classical circuits. Our result improves upon the recent work of Bartusek, Kitagawa, Nishimaki and Yamakawa (STOC 2023) who also showed how to obfuscate pseudo-deterministic quantum circuits in the classical oracle model, but only ones with a completely classical description. Furthermore, our result answers a question of Coladangelo and Gunn, who provide a construction of quantum state indistinguishability obfuscation with respect to a quantum oracle, but leave the existence of a concrete real-world candidate as an open problem. Indeed, our quantum state obfuscator together with Coladangelo-Gunn gives the first candidate realization of a “best-possible” copy-protection scheme for all polynomial-time functionalities. Our techniques deviate significantly from previous works on quantum obfuscation. We develop several novel technical tools which we expect to be broadly useful in quantum cryptography. These tools include a publicly-verifiable, linearly-homomorphic quantum authentication scheme with classically-decodable ZX measurements (which we build from coset states), and a method for compiling any quantum circuit into a ”linear + measurement” () quantum program: an alternating sequence of CNOT operations and partial ZX measurements.

DOI
10.1145/3618260.3649673

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A major unresolved question in quantum cryptography is whether it is possible to obfuscate arbitrary quantum computation. Indeed, there is much yet to understand about the feasibility of quantum obfuscation even in the classical oracle model, where one is given for free the ability to obfuscate any classical circuit. In this work, we develop a new array of techniques that we use to construct a quantum state obfuscator, a powerful notion formalized recently by Coladangelo and Gunn (arXiv:2311.07794) in their pursuit of better software copy-protection schemes. Quantum state obfuscation refers to the task of compiling a quantum program, consisting of a quantum circuit C with a classical description and an auxiliary quantum state ψ, into a functionally-equivalent obfuscated quantum program that hides as much as possible about C and ψ. We prove the security of our obfuscator when applied to any pseudo-deterministic quantum program, i.e. one that computes a (nearly) deterministic classical input / classical output functionality. Our security proof is with respect to an efficient classical oracle, which may be heuristically instantiated using quantum-secure indistinguishability obfuscation for classical circuits. Our result improves upon the recent work of Bartusek, Kitagawa, Nishimaki and Yamakawa (STOC 2023) who also showed how to obfuscate pseudo-deterministic quantum circuits in the classical oracle model, but only ones with a completely classical description. Furthermore, our result answers a question of Coladangelo and Gunn, who provide a construction of quantum state indistinguishability obfuscation with respect to a quantum oracle, but leave the existence of a concrete real-world candidate as an open problem. Indeed, our quantum state obfuscator together with Coladangelo-Gunn gives the first candidate realization of a “best-possible” copy-protection scheme for all polynomial-time functionalities. Our techniques deviate significantly from previous works on quantum obfuscation. We develop several novel technical tools which we expect to be broadly useful in quantum cryptography. These tools include a publicly-verifiable, linearly-homomorphic quantum authentication scheme with classically-decodable ZX measurements (which we build from coset states), and a method for compiling any quantum circuit into a ”linear + measurement” () quantum program: an alternating sequence of CNOT operations and partial ZX measurements.
A classical obfuscator for quantum circuits is a classical program that, given the classical description of a quantum circuit QQQ, outputs the classical description of a functionally equivalent quantum circuit Q^\hat{Q}Q^​ that hides as much as possible about QQQ. Previously, the only known feasibility result for classical obfuscation of quantum circuits (Bartusek and Malavolta, ITCS 2022) was limited to circuits that always reject. On the other hand, if the obfuscator is allowed to compile the quantum circuit QQQ into a quantum state ∣Q^⟩|\hat{Q}\rangle∣Q^​⟩, there exist feasibility results for obfuscating all pseudo-deterministic quantum circuits (Bartusek, Kitagawa, Nishimaki and Yamakawa, STOC 2023, Bartusek, Brakerski and Vaikuntanathan, STOC 2024), and all unitaries (Huang and Tang, FOCS 2025). We show that (relative to a classical oracle) there exists a classical obfuscator for all pseudo-deterministic quantum circuits. We do this by giving the first construction of a compact quantum fully-homomorphic encryption (QFHE) scheme that supports public verification of (pseudo-deterministic) quantum evaluation, relative to a classical oracle. To construct our QFHE scheme, we improve on the approach of Bartusek, Kitagawa, Nishimaki and Yamakawa (STOC 2023), which required ciphertexts that are both quantum and non-compact due to the use of quantum coset states and their publicly-verifiable properties. We introduce new techniques for analyzing coset states that can be generated ''on the fly'', by proving new cryptographic properties of the one-shot signature scheme of Shmueli and Zhandry (CRYPTO 2025). Our techniques allow us to produce QFHE ciphertexts that are purely classical, compact, and publicly-verifiable. This also yields the first classical verification of quantum computation protocol for BQP that simultaneously satisfies blindness and public-verifiability.
Program obfuscation aims to conceal a computer program's internal structure while preserving its functionality. A central open problem is whether an obfuscation scheme for arbitrary quantum circuits exists. Despite several efforts toward this goal, prior works have succeeded only in obfuscating quantum circuits that implement either pseudo-deterministic functions or unitary transformations. Although unitary transformations already include a broad class of quantum computations, many important quantum tasks, such as state preparation, quantum sampling, and quantum error correction, go beyond unitaries and are described by general completely positive trace-preserving (CPTP) maps. In this work, we construct the first quantum ideal obfuscation scheme for arbitrary quantum circuits that support quantum inputs and outputs in the classical oracle model assuming post-quantum one-way functions, thereby resolving an open problem raised in Bartusek et al. (STOC 2023), Bartusek, Brakerski, and Vaikuntanathan (STOC 2024), and Huang and Tang (FOCS 2025). At the core of our construction lies a novel primitive that we introduce, called the subspace-preserving strong pseudorandom unitary (spsPRU). An spsPRU is a family of efficient unitaries that fix every vector in a given linear subspace SSS, while acting as a Haar random unitary on its orthogonal complement S⊥S^\perpS⊥ under both forward and inverse oracle queries. Furthermore, by instantiating the classical oracle model with the ideal obfuscation scheme for classical circuits proposed by Jain et al. (CRYPTO 2023) and later enhanced by Bartusek et al. (CRYPTO 2026), our obfuscation scheme can also be realized in the quantumly accessible pseudorandom oracle model.
Program obfuscation aims to hide the inner workings of a program while preserving its functionality. In the quantum setting, recent works have obtained obfuscation schemes for specialized classes of quantum circuits. For instance, Bartusek, Brakerski, and Vaikuntanathan (STOC 2024) constructed a quantum state obfuscation scheme, which supports the obfuscation of quantum programs represented as quantum states for pseudo-deterministic quantum programs with classical inputs and outputs in the classical oracle model. In this work, we improve upon existing results by constructing the first quantum state obfuscation scheme for unitary (or approximately unitary) quantum programs supporting quantum inputs and outputs in the classical oracle model. At the core of our obfuscation scheme are two novel ingredients: a functional quantum authentication scheme that allows key holders to learn specific functions of the authenticated quantum state with simulationbased security, and a compiler that represents an arbitrary quantum circuit as a projective linear-plus-measurement quantum program described by a sequence of non-adaptive Clifford gates interleaved with adaptive and compatible measurements.
Quantum copy-protection uses the unclonability of quantum states to construct quantum software that provably cannot be pirated. copy-protection would be immensely useful, but unfortunately, little is known about achieving it in general. In this work, we make progress on this goal, by giving the following results: We show how to copy-protect any program that cannot be learned from its input-output behavior relative to a classical oracle. This construction improves on Aaronson (CCC 2009), which achieves the same relative to a quantum oracle. By instantiating the oracle with post-quantum candidate obfuscation schemes, we obtain a heuristic construction of copy-protection. We show, roughly, that any program which can be watermarked can be copy detected, a weaker version of copy-protection that does not prevent copying, but guarantees that any copying can be detected. Our scheme relies on the security of the assumed watermarking, plus the assumed existence of public-key quantum money. Our construction is publicly detectable and applicable to many recent watermarking schemes.
Quantum copy-protection is an innovative idea that uses the no-cloning property of quantum information to copy-protect programs and was first put forward by [Aar09]. The general goal is that a program distributor can distribute a quantum state ∣Ψ⟩|\Psi\rangle∣Ψ⟩, whose classical description is secret to the users; a user can use this state to run the program P on his own input, but not be able to pirate this program P or create another state with the same functionality. In the copy-protection with oracle setting, the user has access to a public oracle and can use the given quantum state and the oracle to compute on his/her own input for polynomially many times. However, the user is not able to produce an additional program(quantum or classical) that computes the same as P on almost all inputs. We present a first quantum copy protection scheme with a classical oracle for any unlearnable function families. The construction is based on membership oracles for hidden subspaces in F2n\mathbb{F}_2^nF2n​, an idea derived from the public key quantum money scheme in[Aar12]. We prove the security of the scheme relative to a classical oracle, namely, the subspace membership oracle with the functionality of computing the secret function we want to copy-protect. The security proof builds on the quantum lower bound for the Direct-Product problem ([Aar12],[BDS16]) and the unlearnability of the copy-protected functions. We also show that existence of quantum copy protection and the quantum hardness of Learning-with-Errors (LWE) will imply publicly verifiable quantum money.
An indistinguishability obfuscator is a probabilistic polynomial-time algorithm that takes a circuit as input and outputs a new circuit that has the same functionality as the input circuit, such that for any two circuits of the same size that compute the same function, the outputs of the indistinguishability obfuscator are computationally indistinguishable. Here, we study schemes for indistinguishability obfuscation for quantum circuits (qiOqi\mathcal{O}qiO), which consists in a procedure that takes as input a quantum circuit and outputs a quantum state, together with a quantum circuit, which together can be used to evaluate the original quantum circuit on any quantum input; the security guarantee being that for two quantum circuits of the same size that have the same input-output behaviour, the outputs of the obfuscator are computationally indistinguishable. Our main result provides a construction for qiOqi\mathcal{O}qiO, where the size of the output of the obfuscator is exponential in the number of non-Clifford (T gates), which means that the construction is efficient as long as the number of T gates is logarithmic in the circuit size.