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Exact Non-Identity Check and Gate-Teleportation-Based Indistinguishability Obfuscation are NP-hard for Low-T-Depth Quantum Circuits

Joshua Nevin

TL;DR

This work proves that Exact Non-Identity Check (ENIC) is NP-hard for Clifford+$\textsf{T}$ circuits with $T$-depth $O(\log n)$, ruling out efficient ENIC and gate-teleportation-based indistinguishability obfuscation (iO) for this circuit class unless $\textsf{P}=\textsf{NP}$. The authors build a deep connection between Pauli-coefficient expansions under Clifford conjugation and depth-$d$ presentations, relating these quantum objects to binary weight enumerator polynomials of linear codes. They develop a cascade of reductions among ENIC, COMMUTE, and related decision problems, and fuse coding-theory hardness (e.g., #P and NP-hard weight-distribution tasks) with quantum circuit analysis via 1-remainder matrices and weight-encoding gadgets. The results reveal fundamental complexity barriers to efficient quantum indistinguishability obfuscation in low-$T$-depth Clifford+$\textsf{T}$ circuits and illuminate rich intersections between quantum circuit analysis and binary coding theory, with implications for cryptographic primitives based on iO/qiO.

Abstract

In 2021, Broadbent and Kazmi developed a gate-teleportation-based protocol for computational indistinguishability obfuscation of quantum circuits. This protocol is efficient for Clifford+T circuits with logarithmically many T-gates, where the limiting factor in the efficiency of the protocol is the difficulty, on input a quantum circuit $C$, of the classical task of producing a description of the unitary obtained by conjugating a Pauli $P$ (corresponding to a Bell-measurement outcome) by $C$, where this description only depends on the input-output functionality of $CPC^{\dagger}$. The task above, in turn, is at least as hard as the problem of determining whether two $n$-qubit quantum circuits are perfectly equivalent up to global phase. In 2009, Tanaka defined the corresponding decision problem Exact Non-Identity Check (ENIC) and showed that ENIC is NQP-complete in general. Motivated by this, we consider in this work what happens when we pass from low T-count to low T-depth. In particular, we show that, for Clifford+T circuits of T-depth $O(\log(n))$, deciding ENIC is NP-hard. This effectively rules out the possibility, for Clifford+T circuits of logarithmic T-depth, of either efficient ENIC or efficient gate-teleportation based computational indistinguishability obfuscation, unless P=NP.

Exact Non-Identity Check and Gate-Teleportation-Based Indistinguishability Obfuscation are NP-hard for Low-T-Depth Quantum Circuits

TL;DR

This work proves that Exact Non-Identity Check (ENIC) is NP-hard for Clifford+ circuits with -depth , ruling out efficient ENIC and gate-teleportation-based indistinguishability obfuscation (iO) for this circuit class unless . The authors build a deep connection between Pauli-coefficient expansions under Clifford conjugation and depth- presentations, relating these quantum objects to binary weight enumerator polynomials of linear codes. They develop a cascade of reductions among ENIC, COMMUTE, and related decision problems, and fuse coding-theory hardness (e.g., #P and NP-hard weight-distribution tasks) with quantum circuit analysis via 1-remainder matrices and weight-encoding gadgets. The results reveal fundamental complexity barriers to efficient quantum indistinguishability obfuscation in low--depth Clifford+ circuits and illuminate rich intersections between quantum circuit analysis and binary coding theory, with implications for cryptographic primitives based on iO/qiO.

Abstract

In 2021, Broadbent and Kazmi developed a gate-teleportation-based protocol for computational indistinguishability obfuscation of quantum circuits. This protocol is efficient for Clifford+T circuits with logarithmically many T-gates, where the limiting factor in the efficiency of the protocol is the difficulty, on input a quantum circuit , of the classical task of producing a description of the unitary obtained by conjugating a Pauli (corresponding to a Bell-measurement outcome) by , where this description only depends on the input-output functionality of . The task above, in turn, is at least as hard as the problem of determining whether two -qubit quantum circuits are perfectly equivalent up to global phase. In 2009, Tanaka defined the corresponding decision problem Exact Non-Identity Check (ENIC) and showed that ENIC is NQP-complete in general. Motivated by this, we consider in this work what happens when we pass from low T-count to low T-depth. In particular, we show that, for Clifford+T circuits of T-depth , deciding ENIC is NP-hard. This effectively rules out the possibility, for Clifford+T circuits of logarithmic T-depth, of either efficient ENIC or efficient gate-teleportation based computational indistinguishability obfuscation, unless P=NP.

Paper Structure

This paper contains 17 sections, 31 theorems, 65 equations, 4 figures.

Table of Contents

  1. Preliminaries: Quantum Gates and Circuits
  2. Obfuscation
  3. Gate Teleportation and Conjugate Encoding
  4. The Main Result and an Overview of the Proof
  5. Cliffords as Symplectomorphisms
  6. Reductions to and from ENIC
  7. Encoding Linear Combinations of Pauli Coefficients of PC-Circuits
  8. Depth-$d$ Presentations
  9. The Coefficient Space of $U^{\Lambda}$
  10. Keeping track of phases
  11. A Formula for $U^{\Lambda}$
  12. Embedding Length-$n$ Codes in $\mathbb{F}_2^{2n}$
  13. Coding Theory Preliminaries
  14. Codes generated by 1-remainder matrices
  15. Relating Coefficients of $U^{\Lambda}$ to Weight Enumerator Polynomials
  16. ...and 2 more sections

Key Result

Theorem 2.5

(GoldwasserRothblumBestPoss, 2007) If the family of 3-CNF formulas admits a (not necessarily efficient) statistical indisitnguishability obfuscator, then the polynomial hierarchy collapses to the second level.

Figures (4)

  • Figure 3.1: The teleportation gadget acting on a $3n$-qubit system
  • Figure 6.1: Entangling $C$ with $2n$ auxiliary registers
  • Figure 6.2: The circuit $V$ detects the presence of $\textsf{P}^z$ in the Pauli expansion of $C$
  • Figure 11.1: Reduction from BINARY-WEIGHT to ENIC and COMMUTE over $\mathcal{T}^{O(\log(n))}$

Theorems & Definitions (75)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 65 more