Magic state distillation with permutation-invariant codes and a two-qubit example

TL;DR

A distillation protocol based on permutation-invariant gnu codes, as small as two qubits, that achieves a 0.5 error threshold and 1/2 distillation rate, surpassing prior schemes for comparable codes.

Abstract

Magic states, by allowing non-Clifford gates through gate teleportation, are important building blocks of fault-tolerant quantum computation. Magic state distillation protocols aim to create clean copies of magic states from many noisier copies. However, the prevailing protocols require substantial qubit overhead. We present a distillation protocol based on permutation-invariant gnu codes, as small as two qubits. The two-qubit protocol achieves a 0.5 error threshold and 1/2 distillation rate, surpassing prior schemes for comparable codes. Our protocol furthermore distils magic states with arbitrary magic by varying the position of the ideal input states on the Bloch sphere. We achieve this by departing from the usual magic state distillation formalism, allowing the use of non-Clifford gates in the distillation protocol, and allowing the form of the output state to differ from the input state. Our protocol is compatible for use in tandem with existing magic state distillation protocols to enhance their performance.

Figures (4)

• Figure 1: (a) A comparison of error thresholds and distillation rates between our distillation protocol and prior work. For comparable small codes, our protocol achieves the highest error threshold and distillation rate. (b) The complete circuit of our distillation protocol as described in Sec. \ref{['sec:distillationprotocol']} using a 2-qubit gnu code with logical states $|0_{1,1,2}\rangle = |00\rangle$ and $|1_{1,1,2}\rangle = ( |10\rangle+|01\rangle)/\sqrt{2}$. (c) Assuming a noiseless channel, the magic of the distilled state is shown as a function of the initial state parameter $v$ with $\theta = \pi/4$ for a $2,3$ and $4$-qubit code with final state given by eqs. \ref{['eq:abc2']}, \ref{['eq:abc3']} and \ref{['eq:abc4']} respectively and magic measured by the 2-Rényi entropy in eq. \ref{['eq:magicdensitymat']}. The gray dashed and dot-dashed lines show the 2-Rényi entropy of the states $|T\rangle$ and $|H\rangle$ respectively. Notice that, with an appropriate choice of initial state and gnu code, states with any amount of magic between $0$ and $M_2(|T\rangle)$ can be distilled using our protocol, even for small codes.
• Figure 2: (a) Initial state parameters and their corresponding magic values required to produce the target states $X|T\rangle\langle T|X$ or $X|H\rangle\langle H|X$ for $2,3$ and $4$-qubit gnu codes. (b) Maximum total output error (see eq. \ref{['eq:maxerror']}) for $2,3$ and $4$-qubit gnu codes (blue) for parameters presented in table (a) to achieve the final state $X|T\rangle\langle T|X$, compared to the output error demonstrated by Bravyi and Kitaev bravyi2005universal (red), see eq. \ref{['eq:paperTerror']}. (c) Maximum total output error for $2,3$ and $4$-qubit gnu codes (blue) for parameters presented in table (a) to achieve the final state $X|H\rangle\langle H|X$, compared to the output error achieved by Bravyi and Kitaev bravyi2005universal (red), see eq. \ref{['eq:paperHerror']}.
• Figure 3: (a) We combine our protocol (labelled $A$) with existing magic state distillation protocol (labelled $B$): Protocol A produces approximate copies of the states $|T\rangle$ or $|H\rangle$ which are then used as inputs in protocol $B$ to create an even higher fidelity $|T\rangle$ or $|H\rangle$. (b) Error suppression of the combined distillation protocol in (a) for $|T\rangle$ and $|H\rangle$ (solid cyan and magenta lines, respectively) where protocol $B$ is chosen to be Bravyi and Kitaev's protocols bravyi2005universal (see eqs. \ref{['eq:paperTerror']} and \ref{['eq:paperHerror']} for definitions of $\epsilon_{out}$ for $|T\rangle$ and $|H\rangle$), compared with a single iteration of protocol $B$ (dot-dashed lines). The combined protocol yields improved error suppression and higher thresholds, from approximately $0.173$ to $0.279$ for $|T\rangle$ and approximately $0.141$ to $0.198$ for $|H\rangle$.
• Figure 4: Maximum error (see Eq. \ref{['eq:maxerror']}) for the 2-qubit repetition code, a special case of the gnu code with $g=2$ and $n=u=1$. The error is shown for initial states defined by the parameters given in Eq. \ref{['eq:Trep']} (black curve) and Eq. \ref{['eq:Hrep']} (blue curve). These initial states (see Eqs. \ref{['eq:initialstates']} and \ref{['eq:initial']}) enable the distillation of $|T\rangle$ and $|H\rangle$ respectively, but provide no error suppression.