Obfuscation of Arbitrary Quantum Circuits (via Subspace-Preserving Pseudorandom Unitary)

Program obfuscation aims to conceal a program’s internal structure while preserving its functionality, and this work constructs the first quantum ideal obfuscation scheme for arbitrary quantum circuits in the classical oracle model. The gist: There exists a quantum obfuscation scheme for arbitrary quantum circuits supporting quantum inputs and outputs, achieving the notion of ideal obfuscation (Definition 4.5) in the classical oracle model assuming postquantum one-way functions.

Challenge of Obfuscating Arbitrary Quantum Circuits

The paper addresses the open problem of whether an obfuscation scheme for arbitrary quantum circuits exists, noting that prior works have only succeeded for pseudo-deterministic functions or unitary transformations, while many important tasks like state preparation and quantum error-correction go beyond unitaries. The two main approaches to constructing quantum obfuscations are restricted: the first approach employs fully homomorphic encryption (QFHE) and classical verification of quantum computation (CVQC), which is fundamentally restricted to circuits whose inputs and outputs are classical. The second approach relies on authenticated quantum codes that support homomorphic computation, achieving obfuscation for any unitary transformation but failing for general completely positive trace-preserving maps because the auxiliary state may become entangled with the output, preventing coherent restoration.

Our Solution: Subspace-Preserving Strong Pseudorandom Unitary (spsPRU)

The solution builds upon the obfuscation scheme of [HT25] for unitary circuits by transforming an arbitrary quantum circuit Q implementing a general quantum map Φ into a circuit Q' with unitary functionality that hides Q but allows the computation of Φ. The core primitive introduced is the subspace-preserving strong pseudorandom unitary (spsPRU). An spsPRU is defined as a family of efficient unitaries that fix every vector in a given linear subspace S, while acting as a Haar random unitary on the orthogonal complement S⊥ under both forward and inverse oracle queries. This primitive is crucial because it exhibits both structure (in S) and pseudorandomness (in S⊥), allowing the construction to ensure that "the input and ancilla remain the same state conditioned that the ancilla is in the all 1⟩ state, while fully scrambling the inputs and ancilla into a pseudorandom state (in S⊥) conditioned that the ancilla is incorrectly prepared."

The Ideal Obfuscation Scheme

The paper defines ideal obfuscation (Definition 4.5) for arbitrary quantum maps by requiring that anything an adversary learns from the obfuscated circuit is computationally indistinguishable to what the adversary learns from interacting with an ideal functionality Idealλ,p,Φ which depends only on the security parameter λ, circuit parameter p, and the quantum map Φ being implemented. The authors propose a unique probability measure µunifΦ on the set of unitary extensions U(2m)Φ that captures this computational power. The final construction (Construction 2) utilizes an ideal obfuscation scheme for Qunitary circuits (Theorem 4.6), a strong pseudorandom unitary PRU, and the subspace-preserving spsPRU to construct the final obfuscated program Q′k,k' = (PRUk′ ⊗ In′) ◦ (ctrl1λ-UQ) ◦ spsPRU.

Construction Details and Security Analysis

The construction involves several steps parameterized by a security parameter λ:

1. Increasing the number of working qubits to m' = λ + m and setting the dimension d = 2m' − 2n.

2. Setting the subspace S as span x⟩ for x ∈ [d].

3. Forming the unitary circuit Q′k,k′ using a random key k for spsPRU and a random key k' for the (m'-n')-qubit PRU, incorporating a controlled unitary operation (ctrl1λ-UQ).

4. The evaluation algorithm QEvalF(ρin, ψe) recovers the last n' qubits of the output state ρ'.

The security proof demonstrates that for any two circuits Q0 and Q1 with close mappings Φ0 and Φ1, the resulting obfuscations are indistinguishable against any polynomial query adversary D, showing that Pr W0←µunifΦ0 h1 ← DW0,W†0 i − Pr W1←µunifΦ1 h1 ← DW1,W†1 i = negl(λ). This establishes that the scheme is both ideal obfuscation and indistinguishability obfuscation. The efficiency of the construction is maintained because spsPRU is poly(n)-time computable within an inverse exponential error in diamond distance for all k ∈ K.

Ideal Functionality Implementation

The ideal functionality Idealλ,p,Φ is efficiently implemented using the S-preserving path-recording oracle Orecord and O'record. These oracles are defined by extending the path-recording oracle [MH25] to preserve the subspace S = span x⟩ for x ∈ [d].

Based on the provided scientific paper, Obfuscation of Arbitrary Quantum Circuits, here are specific improvements for AI systems that leverage these concepts, along with what those improved systems could achieve:

The paper introduces a method for constructing an ideal obfuscation scheme for arbitrary quantum circuits in the classical oracle model. This scheme relies on a novel primitive called the subspace-preserving strong pseudorandom unitary (spsPRU).

Here are the specific improvements and capabilities derived from this research:

1. Improving Quantum Circuit Security via Ideal Obfuscation:

2. Enhancing Robustness Against Structural Attacks:

3. Achieving General Functionality Obfuscation (Beyond Unitaries):

4. Enabling Watermarking and AI Program Protection:

Specific Improvements and Capabilities of the Improved AI System:

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 S, while acting as a Haar random unitary on its orthogonal complement S 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.