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Adaptive Quantum Homeopathy

Lennart Bittel, Lorenzo Leone

TL;DR

Compared to existing approaches, this method significantly reduces non-Clifford overhead while strengthening security guarantees to adaptive security as well as removing artificial assumptions between $n and $k$.

Abstract

Randomness is a fundamental resource in quantum information, with crucial applications in cryptography, algorithms, and error correction. A central challenge is to construct unitary $k$-designs that closely approximate Haar-random unitaries while minimizing the costly use of non-Clifford operations. In this work, we present a protocol, named Quantum Homeopathy, able to generate unitary $k$-designs on $n$ qubits, secure against any adversarial quantum measurement, with a system-size-independent number of non-Clifford gates. Inspired by the principle of homeopathy, our method applies a $k$-design only to a subsystem of size $Θ(k)$, independent of $n$. This "seed" design is then "diluted" across the entire $n$-qubit system by sandwiching it between two random Clifford operators. The resulting ensemble forms an $\varepsilon$-approximate unitary $k$-design on $n$ qubits. We prove that this construction achieves full quantum security against adaptive adversaries using only $\tilde{O}(k^2 \log\varepsilon^{-1})$ non-Clifford gates. If one requires security only against polynomial-time adaptive adversaries, the non-Clifford cost decreases to $\tilde{O}(k + \log^{1+c} \varepsilon^{-1})$. This is optimal, since we show that at least $Ω(k)$ non-Clifford gates are required in this setting. Compared to existing approaches, our method significantly reduces non-Clifford overhead while strengthening security guarantees to adaptive security as well as removing artificial assumptions between $n$ and $k$. These results make high-order unitary designs practically attainable in near-term fault-tolerant quantum architectures.

Adaptive Quantum Homeopathy

TL;DR

Compared to existing approaches, this method significantly reduces non-Clifford overhead while strengthening security guarantees to adaptive security as well as removing artificial assumptions between k$.

Abstract

Randomness is a fundamental resource in quantum information, with crucial applications in cryptography, algorithms, and error correction. A central challenge is to construct unitary -designs that closely approximate Haar-random unitaries while minimizing the costly use of non-Clifford operations. In this work, we present a protocol, named Quantum Homeopathy, able to generate unitary -designs on qubits, secure against any adversarial quantum measurement, with a system-size-independent number of non-Clifford gates. Inspired by the principle of homeopathy, our method applies a -design only to a subsystem of size , independent of . This "seed" design is then "diluted" across the entire -qubit system by sandwiching it between two random Clifford operators. The resulting ensemble forms an -approximate unitary -design on qubits. We prove that this construction achieves full quantum security against adaptive adversaries using only non-Clifford gates. If one requires security only against polynomial-time adaptive adversaries, the non-Clifford cost decreases to . This is optimal, since we show that at least non-Clifford gates are required in this setting. Compared to existing approaches, our method significantly reduces non-Clifford overhead while strengthening security guarantees to adaptive security as well as removing artificial assumptions between and . These results make high-order unitary designs practically attainable in near-term fault-tolerant quantum architectures.

Paper Structure

This paper contains 10 sections, 31 theorems, 74 equations, 2 figures, 1 table.

Table of Contents

  1. Overview of previous results
  2. Preliminaries
  3. Notation
  4. Pauli and Clifford group
  5. Unitary $k$-designs
  6. Haar average over the Clifford group
  7. Proof of the main results
  8. Adaptive security: proof of \ref{['maintheorem']}
  9. Lower bounds on the non-Clifford cost
  10. Implications of our results

Key Result

Theorem 1

The ensemble of unitaries $\mathcal{E}_t$ forms a $\varepsilon$-approximate quantum-secure unitary $k$-design provided that $t\ge 2k+\log\varepsilon^{-1}+6$.

Figures (2)

  • Figure 1: Clifford circuit (orange boxes) with few interspersed non-Clifford gates (green squares).
  • Figure 2: Quantum homeopathy: mixing power of Clifford circuits spreads the design "seed" into the full $n$-qubit space. The figure provides a graphical representation of \ref{['maintheorem']}.

Theorems & Definitions (62)

  • Definition 1: Quantum-secure unitary designs
  • Definition 2: Quantum-polynomially-secure unitary $k$-designs
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Corollary 3
  • Definition 3: Clifford group
  • Definition 4: Stabilizer states
  • Definition 5: $k$-fold channel
  • ...and 52 more