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Transversal AND in Quantum Codes

Christine Li, Lia Yeh

Abstract

The AND gate is not reversible$\unicode{x2014}$on qubits. However, it is reversible on qutrits, making it a building block for efficient simulation of qubit computation using qutrits. We first observe that there are multiple two-qutrit Clifford+T unitaries that realize the AND gate with T-count 3, and its generalizations to $n$ qubits with T-count $3n-3$. Our main result is the construction of a novel qutrit $\mathopen{[\![} 6,2,2 \mathclose{]\!]}$ quantum error-correcting code with a transversal implementation of the AND gate. The key insight in our approach is that a symmetric T-depth one circuit decomposition$\unicode{x2014}$composed of a CX circuit, T and T dagger gates, followed by the CX circuit in reverse$\unicode{x2014}$of a given unitary can be interpreted as a CSS code. We can increase the code distance by augmenting the code circuit with additional stabilizers while preserving the logical gate. This results in a code with a "built-in" transversal implementation of the original unitary, which can be further concatenated to attain a $\mathopen{[\![} 48,2,4 \mathclose{]\!]}$ code with the same transversal logical gate. Furthermore, we present several protocols for mixed qubit-qutrit codes which we call Qubit Subspace Codes, and for magic state distillation and injection.

Transversal AND in Quantum Codes

Abstract

The AND gate is not reversibleon qubits. However, it is reversible on qutrits, making it a building block for efficient simulation of qubit computation using qutrits. We first observe that there are multiple two-qutrit Clifford+T unitaries that realize the AND gate with T-count 3, and its generalizations to qubits with T-count . Our main result is the construction of a novel qutrit quantum error-correcting code with a transversal implementation of the AND gate. The key insight in our approach is that a symmetric T-depth one circuit decompositioncomposed of a CX circuit, T and T dagger gates, followed by the CX circuit in reverseof a given unitary can be interpreted as a CSS code. We can increase the code distance by augmenting the code circuit with additional stabilizers while preserving the logical gate. This results in a code with a "built-in" transversal implementation of the original unitary, which can be further concatenated to attain a code with the same transversal logical gate. Furthermore, we present several protocols for mixed qubit-qutrit codes which we call Qubit Subspace Codes, and for magic state distillation and injection.
Paper Structure (31 sections, 19 theorems, 63 equations, 2 figures, 2 tables)

This paper contains 31 sections, 19 theorems, 63 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Qubit and Qutrit Reversible Computation
  3. Qubit and Qutrit ZX-calculus
  4. The AND Gate as a Quantum Circuit Primitive
  5. Emulation: Qubit Computation using Qutrits
  6. A T-depth 1 Symmetric Circuit as a [[3,2,1]] Qutrit Code
  7. Reconstructing the Code at Higher Distance
  8. Recovering Logical AND
  9. Deriving the Code as a Symmetric, T-depth One Circuit
  10. Finding the Stabilizers and Logical Operators
  11. Getting to Distance 4 Through Code Concatenation
  12. Qubit Subspace Codes
  13. Logical Qubit, Physical Qutrits
  14. Qubit Subspace Projection
  15. Magic State Distillation and Injection
  16. ...and 16 more sections

Key Result

lemma 1

The binary AND gate can be emulated by a two-qutrit Clifford+T unitary with T-count 3.

Figures (2)

  • Figure 1: Rewrite rules for the qutrit ZX-calculus, reprinted from Figure 1 in townsend-teague_simplification_2022
  • Figure 2: Figure for a T-depth one decomposition of the qubit CCZ gate, reprinted from SelingerP2013tdepthone.

Theorems & Definitions (43)

  • definition 1: Qutrit X gates
  • definition 2: Qutrit X basis
  • definition 3: Hadamard gate
  • definition 4: Z phase gate
  • definition 5: X phase gate
  • definition 6: T gate
  • definition 7: $\Lambda$-controlled $U$
  • definition 8: $\ket{0}$-controlled $U$ gate
  • lemma 1
  • proof
  • ...and 33 more