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Non-Clifford Gates are Required for Long-Term Memory

Jon Nelson, Joel Rajakumar, Michael J. Gullans

TL;DR

The paper proves a no-go theorem: Clifford circuits with mid-circuit resets and constant-rate depolarizing noise cannot preserve quantum information for long-term memory. By recasting the dynamics in a Pauli-path framework and using adjoint maps to handle non-unital reset channels, the authors bound the survival of non-identity Pauli operators and show that after $d^* = O(\gamma^{-1}\log(n)\log(n/\epsilon))$ layers the output states become indistinguishable: $||\Phi(\rho)-\Phi(\sigma)||_1 \le \epsilon$. The main technical device is a batched counting argument that halves the effective Pauli weight per batch, leading to a $d = O(\log^2 n)$ total depth necessary to kill all Pauli paths. This demonstrates that non-Clifford gates are fundamentally required for fault-tolerant memory, even with resets and fresh qubits, highlighting the operational role of magic in preserving information under noise.

Abstract

We show that all Clifford circuits under interspersed depolarizing noise lose memory of their input exponentially quickly, even when given access to a constant supply of fresh qubits in arbitrary states. This is somewhat surprising given the result of Aharonov et al. [STOC1997] which gives a fault-tolerant protocol for general quantum circuits using a supply of fresh qubits. Our result shows that such a protocol is impossible using only Clifford gates demonstrating that non-Clifford gates are fundamentally required to store information for long periods of time.

Non-Clifford Gates are Required for Long-Term Memory

TL;DR

The paper proves a no-go theorem: Clifford circuits with mid-circuit resets and constant-rate depolarizing noise cannot preserve quantum information for long-term memory. By recasting the dynamics in a Pauli-path framework and using adjoint maps to handle non-unital reset channels, the authors bound the survival of non-identity Pauli operators and show that after layers the output states become indistinguishable: . The main technical device is a batched counting argument that halves the effective Pauli weight per batch, leading to a total depth necessary to kill all Pauli paths. This demonstrates that non-Clifford gates are fundamentally required for fault-tolerant memory, even with resets and fresh qubits, highlighting the operational role of magic in preserving information under noise.

Abstract

We show that all Clifford circuits under interspersed depolarizing noise lose memory of their input exponentially quickly, even when given access to a constant supply of fresh qubits in arbitrary states. This is somewhat surprising given the result of Aharonov et al. [STOC1997] which gives a fault-tolerant protocol for general quantum circuits using a supply of fresh qubits. Our result shows that such a protocol is impossible using only Clifford gates demonstrating that non-Clifford gates are fundamentally required to store information for long periods of time.

Paper Structure

This paper contains 13 sections, 3 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Implications
  3. Proof Strategy
  4. Open Questions
  5. Background and Notation
  6. Pauli and Clifford Group
  7. Stochastic Noise
  8. Reset Channel
  9. Adjoint Maps
  10. Analysis
  11. Reduction to Pauli Path Survival Probability
  12. Grouping Pauli Operators based on Weight
  13. Batched Counting Argument

Key Result

theorem 1.1

Let $\Phi$ denote the channel produced by a circuit of $d$ layers of Clifford gates and reset gates on $n$ qubits, with single-qubit depolarizing noise of strength $\gamma$ on each qubit after each layer. Let $\rho$ and $\sigma$ be two arbitrary input states. When $d > d^*$ where $d^* = O(\gamma^{-1

Theorems & Definitions (6)

  • theorem 1.1
  • lemma 3.1
  • proof
  • definition 3.2
  • lemma 3.4
  • proof