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arXiv:1210.3388 · Physical Review A

Multilevel distillation of magic states for quantum computing

Cody Jones

Published Apr 4, 2013• 126 citations
arXivHTMLPDF

Abstract

We develop a procedure for distilling magic states used in universal quantum computing that requires substantially fewer initial resources than prior schemes. Our distillation circuit is based on a family of concatenated quantum codes that possess a transversal Hadamard operation, enabling each of these codes to distill the eigenstate of the Hadamard operator. A crucial result of this design is that low-fidelity magic states can be consumed to purify other high-fidelity magic states to even higher fidelity, which we call multilevel distillation. When distilling in the asymptotic regime of infidelity $\ensuremath{\epsilon}\ensuremath{\rightarrow}0$ for each input magic state, the number of input magic states consumed on average to yield an output state with infidelity $O({\ensuremath{\epsilon}}^{{2}^{r}})$ approaches ${2}^{r}+1$, which comes close to saturating the conjectured bound in another investigation [Bravyi and Haah, Phys. Rev. A 86, 052329 (2012)]. We show numerically that there exist multilevel protocols such that the average number of magic states consumed to distill from error rate ${\ensuremath{\epsilon}}_{\mathrm{in}}=0.01$ to ${\ensuremath{\epsilon}}_{\mathrm{out}}$ in the range ${10}^{\ensuremath{-}5}$--${10}^{\ensuremath{-}40}$ is about $14{\mathrm{log}}_{10}(1/{\ensuremath{\epsilon}}_{\mathrm{out}})\ensuremath{-}40$; the efficiency of multilevel distillation dominates all other reported protocols when distilling Hadamard magic states from initial infidelity 0.01 to any final infidelity below ${10}^{\ensuremath{-}7}$. These methods are an important advance for magic-state distillation circuits in high-performance quantum computing and provide insight into the limitations of nearly resource-optimal quantum error correction.

DOI
10.1103/physreva.87.042305

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We develop a procedure for distilling magic states used in universal quantum computing that requires substantially fewer initial resources than prior schemes. Our distillation circuit is based on a family of concatenated quantum codes that possess a transversal Hadamard operation, enabling each of these codes to distill the eigenstate of the Hadamard operator. A crucial result of this design is that low-fidelity magic states can be consumed to purify other high-fidelity magic states to even higher fidelity, which we call multilevel distillation. When distilling in the asymptotic regime of infidelity \ensuremathϵ\ensuremath→0\ensuremath{\epsilon}\ensuremath{\rightarrow}0\ensuremathϵ\ensuremath→0 for each input magic state, the number of input magic states consumed on average to yield an output state with infidelity O(\ensuremathϵ2r)O({\ensuremath{\epsilon}}^{{2}^{r}})O(\ensuremathϵ2r) approaches 2r+1{2}^{r}+12r+1, which comes close to saturating the conjectured bound in another investigation [Bravyi and Haah, Phys. Rev. A 86, 052329 (2012)]. We show numerically that there exist multilevel protocols such that the average number of magic states consumed to distill from error rate \ensuremathϵin=0.01{\ensuremath{\epsilon}}_{\mathrm{in}}=0.01\ensuremathϵin​=0.01 to \ensuremathϵout{\ensuremath{\epsilon}}_{\mathrm{out}}\ensuremathϵout​ in the range 10\ensuremath−5{10}^{\ensuremath{-}5}10\ensuremath−5--10\ensuremath−40{10}^{\ensuremath{-}40}10\ensuremath−40 is about 14log10(1/\ensuremathϵout)\ensuremath−4014{\mathrm{log}}_{10}(1/{\ensuremath{\epsilon}}_{\mathrm{out}})\ensuremath{-}4014log10​(1/\ensuremathϵout​)\ensuremath−40; the efficiency of multilevel distillation dominates all other reported protocols when distilling Hadamard magic states from initial infidelity 0.01 to any final infidelity below 10\ensuremath−7{10}^{\ensuremath{-}7}10\ensuremath−7. These methods are an important advance for magic-state distillation circuits in high-performance quantum computing and provide insight into the limitations of nearly resource-optimal quantum error correction.
Magic state distillation is a crucial yet resource-intensive process in fault-tolerant quantum computation. The protocol's overhead, defined as the number of input magic states required per output magic state with an error rate below εεε, typically grows as O(log⁡γ(1/ε))\mathcal{O}(\log^γ(1/ε))O(logγ(1/ε)). Achieving smaller overheads, i.e., smaller exponents γγγ, is highly desirable; however, all existing protocols require polylogarithmically growing overheads with some γ>0γ> 0γ>0, and identifying the smallest achievable exponent γγγ for distilling magic states of qubits has remained challenging. To address this issue, we develop magic state distillation protocols for qubits with efficient, polynomial-time decoding that achieve an O(1)\mathcal{O}(1)O(1) overhead, meaning the optimal exponent γ=0γ= 0γ=0; this improves over the previous best of γ≈0.678γ\approx 0.678γ≈0.678 due to Hastings and Haah. In our construction, we employ algebraic geometry codes to explicitly present asymptotically good quantum codes for 2102^{10}210-dimensional qudits that support transversally implementable logical gates in the third level of the Clifford hierarchy. The use of asymptotically good codes with non-vanishing rate and relative distance leads to the constant overhead. These codes can be realised by representing each 2102^{10}210-dimensional qudit as a set of 101010 qubits, using stabiliser operations on qubits. The 101010-qubit magic states distilled with these codes can be converted to and from conventional magic states for the controlled-controlled-ZZZ (CCZCCZCCZ) and TTT gates on qubits with only a constant overhead loss, making it possible to achieve constant-overhead distillation of such standard magic states for qubits. These results resolve the fundamental open problem in quantum information theory concerning the construction of magic state distillation protocols with the optimal exponent.
The distillation of magic states is an often-cited technique for enabling universal quantum computing once the error probability for a special subset of gates has been made negligible by other means. We present a routine for magic-state distillation that reduces the required overhead for a range of parameters of practical interest. Each iteration of the routine uses a four-qubit error-detecting code to distill the +1+1+1 eigenstate of the Hadamard gate at a cost of ten input states per two improved output states. Use of this routine in combination with the 151515-to-111 distillation routine described by Bravyi and Kitaev allows for further improvements in overhead.
The distillation of magic states is an often-cited technique for enabling universal quantum computing once the error probability for a special subset of gates has been made negligible by other means. We present a routine for magic-state distillation that reduces the required overhead for a range of parameters of practical interest. Each iteration of the routine uses a four-qubit error-detecting code to distill the +1 eigenstate of the Hadamard gate at a cost of ten input states per two improved output states. Use of this routine in combination with the 15-to-1 distillation routine described by Bravyi and Kitaev allows for further improvements in overhead.
Magic-state distillation is a resource intensive subroutine for quantum computation. The ratio of noisy input states to output states with an error rate at most ε scales as O(log^{γ}(1/ε)) [S. Bravyi and J. Haah, Magic-state distillation with low overhead, Phys. Rev. A 86, 052329 (2012)10.1103/PhysRevA.86.052329]. In a breakthrough paper, Hastings and Haah [Distillation with Sublogarithmic Overhead, Phys. Rev. Lett. 120, 050504 (2018)10.1103/PhysRevLett.120.050504] showed that it is possible to construct distillation routines with a sublogarithmic overhead, achieving γ≈0.6779 and falsifying a conjecture that γ is lower bounded by 1. They then ask whether γ can be made arbitrarily close to 0. We answer this question in the affirmative for magic-state distillation routines using qudits of prime dimension (d dimensional quantum systems for prime d).
Magic state distillation (MSD) is an essential element for universal fault-tolerant quantum computing, which distills a high-fidelity magic state from noisy magic states using ideal (error-corrected) Clifford operations. For ideal Clifford operations, it needs to be performed on the logical qubits and hence incurs a large spatiotemporal overhead, which is one of the major bottlenecks for the realization of fault-tolerant quantum computers (FTQCs). Here we propose zero-level distillation, which prepares a high-fidelity logical magic state at the physical level, namely , using physical qubits and nearest-neighbor two-qubit gates on a square lattice. We develop a zero-level distillation circuit and show that distillation can be made even more efficient than the conventional sophisticated approaches with logical level distillations. The key idea involves the Knill -type distillation using the Steane code and its careful mapping to the square-lattice architecture with error detection. The distilled magic state on the Steane-code state is then teleported or converted to surface codes. We numerically find that the error rate of the logical magic state scales as approximately 100 × p 2 in terms of the physical error rate p . For example, with a physical error rate of p = 10 − 4 ( 10 − 3 ), the logical error rate is reduced to p L = 10 − 6 ( 10 − 4 ), resulting in an improvement of 2 (1) orders of magnitude. This contributes to reducing both space and time overhead for early FTQC as well as full-fledged FTQC combined with conventional multilevel distillation protocols.
Recently [1] we proposed a family of magic state distillation protocols that obtains asymptotic performance that is conjectured to be optimal. This family depends upon several codes, called “inner codes” and “outer codes.” In Ref. [1], some small examples of these codes were given as well as an analysis of codes in the asymptotic limit. Here, we analyze such protocols in an intermediate size regime, using hundreds to thousands of qubits. We use BCH inner codes [2], combined with various outer codes. We extend the protocols of Ref. [1] by adding error correction in some cases. We present a variety of protocols in various input error regimes; in many cases these protocols require significantly fewer input magic states to obtain a given output error than previous protocols.