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Quantum Error Correction by Purification

Jonathan Raghoonanan, Tim Byrnes

Abstract

We present a general-purpose quantum error correction primitive based on state purification via the SWAP test, which we refer to as purification quantum error correction (PQEC). This method operates on $N$ noisy copies, requires minimally $O(M\log_2 N)$ data qubits to process the $M$-qubit inputs. In a similar way to standard QEC, the purification steps may be interleaved within a quantum algorithm to suppress the logical error rate. No postselection is performed and no knowledge of the state is required. We analyze its performance under a variety of error channels and find that PQEC is highly effective at boosting fidelity and reducing logical error rates, particularly for the depolarizing channel. Error thresholds for the local depolarizing channel are found to be $ 75 \%$ for any register size. For local dephasing, the error threshold is reduced to $ 50 \% $ but may be boosted using twirling.

Quantum Error Correction by Purification

Abstract

We present a general-purpose quantum error correction primitive based on state purification via the SWAP test, which we refer to as purification quantum error correction (PQEC). This method operates on noisy copies, requires minimally data qubits to process the -qubit inputs. In a similar way to standard QEC, the purification steps may be interleaved within a quantum algorithm to suppress the logical error rate. No postselection is performed and no knowledge of the state is required. We analyze its performance under a variety of error channels and find that PQEC is highly effective at boosting fidelity and reducing logical error rates, particularly for the depolarizing channel. Error thresholds for the local depolarizing channel are found to be for any register size. For local dephasing, the error threshold is reduced to but may be boosted using twirling.
Paper Structure (39 sections, 1 theorem, 93 equations, 12 figures)

This paper contains 39 sections, 1 theorem, 93 equations, 12 figures.

Table of Contents

  1. Introduction
  2. SWAP-based purification
  3. The SWAP test
  4. SWAP gadget
  5. Purification QEC
  6. Basic idea: Single SWAP purification
  7. Multiple purification rounds
  8. Extracting observables
  9. Purifying quantum circuits
  10. Qubit-efficient PQEC
  11. Equivalent purification circuit
  12. Interleaving quantum algorithms
  13. Error-model independent performance
  14. Noisy single qubit
  15. Purification
  16. ...and 24 more sections

Key Result

Theorem 1

For any $D\ge 2$, the recursion $F'=g(F)$ in Eq. eq:gF-def has fixed points at $F_\star=1$ and $F_\star=1/D$ (and also $F_\star=0$ for the extended recursion on $F\in[0,1]$). Moreover: (i) if $F>1/D$ then $F'$ is strictly increasing and converges to $1$; (ii) $F=1/D$ is a fixed point (the maximally

Figures (12)

  • Figure 1: Purification quantum error correction implementation in a quantum circuit. (a) Depth-efficient configuration of SWAP-based purifiers being interleaved within a $M$-qubit quantum algorithm consisting of a sequence of unitary gates $U_n$. The whole quantum circuit is given by the unitary $U = \prod_n U_n$. The state $\ket{\psi_n}^{(\ell)}$ refers to the state at the $n$th stage of the quantum algorithm, with $\ell$ rounds of purification. (b) The SWAP gadget and its definition. The $\Pi_+$ measurement corresponds to the ancilla outcome $| 0 \rangle$ and $\Pi_-$ corresponds to $| 1 \rangle$. The second copy of the state is discarded (traced out). Thick horizontal lines denote qubit registers consisting of $M$ qubits, thin horizontal lines are single qubits.
  • Figure 2: Qubit-efficient implementation of the PQEC scheme. Purification sequences for (a) $\ell = 1$; (b) $\ell = 2$; (c) $\ell = 3$ are shown. $\cal{P}_\ell$ denotes a circuit that consumes $N=2^\ell$ noisy preparations of $\ket{\psi}^{(0)}$ and outputs a single purified state $\ket{\psi}^{(\ell)}$ that has successfully passed through $\ell$ rounds of SWAP purification. In (a), the two registers are first SWAP purified according to the circuit given in Fig. \ref{['diag:main_use_case']}(b). The discarded register is then reinitialized to the $\ell = 0$ level purified state $| \psi \rangle^{(0)}$.
  • Figure 3: Interleaving SWAP-based purification within a quantum algorithm. (a) Circuit identity for interchanging the order of the purification step ${\cal P}_1$ and an arbitrary unitary $U$. (b) Absorbing a unitary on the reinitialization register of ${\cal P}_1$. (c) Circuit rearrangement for a single unitary gate acting on the $\ell = 2$ purified state. (d) Circuit rearrangement for two unitary gates acting on the $\ell = 3$ purified state. The definition of the block $U_1 {\cal P}_1$ is shown in (c).
  • Figure 4: Output fidelity versus input fidelity after a single pufication round for the Werner channel \ref{['eq:isotropic-family']}, for several register dimensions $D$. Curves show the exact map of Eq. \ref{['eq:gen_error_reduction']} with the gray dashed line indicating the identity $F' = F$. The purification step strictly improves fidelity when $F> \frac{1}{D}$, and the gain is largest at lower $F$ where the curvature is steepest.
  • Figure 5: Error threshold behavior for a single qubit ($D=2$) in a global depolarizing channel with PQEC. (a)(b) Fidelity evolution under the depolarizing Bloch vector contraction (\ref{['eq:isotropic-family']}) and $\ell$ rounds of purification (\ref{['qubitradialupdate']}), where $t$ is the number of cycles. (a) Fidelity evolution versus $t$ with $p = 0.1$ and various $\ell$. (b) Fidelity evolution versus $t$ for $\ell = 1$ and various $p$ as shown. (c) The steady state ($t \rightarrow \infty$) fidelity $F_0$ for various $\ell$. (d) The logical error rate $\gamma_{L}$ evaluated using (\ref{['gammadef']}) for various $\ell$.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Definition 1: Error threshold for PQEC
  • Theorem 1: Monotone behavior and convergence in the isotropic family
  • proof