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Multilevel distillation of magic states for quantum computing

Cody Jones

arXiv:1210.3388v2quant-ph

TL;DR

Magic-state distillation supports universal fault-tolerant quantum computing, but its circuits are estimated to consume most quantum-computer resources. This paper uses concatenated codes with transversal Hadamard for multilevel distillation, achieving strong asymptotic efficiency and outperforming reported protocols below 10^-7 final infidelity.

  • Problem

    Magic-state distillation is important for universal fault-tolerant quantum computing, yet its circuits are estimated to consume most quantum-computer resources.

    Source: §I · p. 1

  • Method

    The authors use CSS H codes with transversal Hadamard and concatenate them so low-fidelity magic-state ancillas can purify higher-fidelity inputs at different levels.

    Source: §I · pp. 1–2, §II · p. 2

  • Results

    As ε approaches zero, r-round protocols use asymptotically 2^r + 1 inputs per output for O(ε^(2^r)) infidelity and outperform reported protocols below 10^-7 final infidelity.

    Source: §I · p. 2, §III · p. 4

  • Takeaways & Limitations

    Multilevel distillation offers an efficient approach for large-scale fault-tolerant computing, where magic-state distillation is often considered the most costly subroutine.

    Source: §V · p. 5

    Multilevel circuits can be larger in qubits and gates, and output errors are correlated, so distilled states must not meet again in a subsequent distillation circuit.

    Source: §III · p. 3

Abstract

from arXiv · show

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 $\epsilon \rightarrow 0$ for each input magic state, the number of input magic states consumed on average to yield an output state with infidelity $O(\epsilon^{2^r})$ approaches $2^r+1$, which comes close to saturating the conjectured bound in [Phys. Rev. A 86, 052329]. We show numerically that there exist multilevel protocols such that the average number of magic states consumed to distill from error rate $\epsilon_{\mathrm{in}} = 0.01$ to $\epsilon_{\mathrm{out}}$ in the range $10^{-5}$ to $10^{-40}$ is about $14\log_{10}(1/\epsilon_{\mathrm{out}}) - 40$; 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 $10^{-7}$. These methods are an important advance for magic-state distillation circuits in high-performance quantum computing, and they provide insight into the limitations of nearly resource-optimal quantum error correction.

I. INTRODUCTION

Magic-state distillation helps achieve universal fault-tolerant quantum computing, but distillation circuits are estimated to consume most quantum-computer resources. This paper introduces transversal-Hadamard H codes and multilevel protocols, which improve distillation efficiency.

I. INTRODUCTION

  • Magic-state distillation supports universal fault-tolerant quantum computing, and its circuits are estimated to require most quantum-computer resources.Magic states are injected into an error-corrected code, purified, and consumed to implement otherwise unavailable circuits.Source: §I · p. 1
  • Prior work includes 10-to-2 and (3k + 8)-to-k distillation procedures, while this work continues the trend toward larger multiqubit codes.Earlier studies also explored topological-code circuit reductions, experimental NMR distillation, and qudit protocols.Source: §I · p. 1
  • H codes are CSS codes with transversal Hadamard, and concatenated H codes enable multilevel distillation using low-fidelity ancillas to purify higher-fidelity magic states.H codes encode n − 4 logical qubits in n physical qubits; the protocols use inputs at different infidelity levels and concatenation levels.Source: §I · pp. 1–2
  • Multilevel protocols yield O(ε^(2^r)) output infidelity using asymptotically 2^r + 1 input states per output, and outperform reported protocols below 10^-7 final infidelity.The asymptotic result assumes suitably small infidelity in each input state.Source: §I · p. 2

II. A FAMILY OF CODES WITH TRANSVERSAL HADAMARD

The paper defines CSS H codes that encode k logical qubits in k + 4 physical qubits and support transversal Hadamard. Encoding faulty |H⟩ states and detecting errors with Hadamard-basis measurements enables distillation.

II. A FAMILY OF CODES WITH TRANSVERSAL HADAMARD

  • H codes are CSS codes with k even, k + 4 physical qubits, distance two, and transversal Hadamard inherited from the physical-qubit Hadamard gates.Distance two means the code detects a single physical Pauli error.Source: §II · p. 2
  • An H_(k+4) code encodes k |H⟩ states, and transversal controlled-Hadamard operations consume two |H⟩ states per physical qubit, giving a (3k + 8)-to-k distillation procedure.The circuit rejects outputs when either the Hadamard measurement or code-stabilizer checks detect an error.Source: §II · p. 2, Figure 1 · p. 2
  • The +1 Hadamard eigenstate |H⟩ is distilled by encoding faulty states, measuring in the transversal-Hadamard basis with |H⟩ ancillas, and rejecting detected errors.Two |H⟩ states implement a controlled-H operation used for the measurement.Source: §II · p. 2

III. MULTILEVEL DISTILLATION

Multilevel distillation uses concatenated codes with transversal Hadamard operations to consume lower-fidelity magic-state ancillas while purifying higher-fidelity encoded states. Its asymptotic resource scaling approaches the conjectured lower bound, though repeated-round circuits can become unwieldy.

III. MULTILEVEL DISTILLATION

  • Concatenated transversal-Hadamard codes use lower-fidelity magic states for controlled-Hadamard measurements while distilling higher-fidelity encoded states; r-level concatenation has distance 2^r.When logical errors are independent, a two-level code suppresses them to O(ε_l^2).Source: §III · pp. 2–3
  • To preserve distance multiplication, any pair of physical qubits may share at most one encoding block across all concatenation levels.The restriction addresses correlated errors within a block and the independence assumption used for multiplying code distances.Source: §III · p. 3
  • For a two-level square code, measurement failures scale as O(ε_p^4)+O(ε_l ε_p^2), while even numbers of encoded errors are not detected.The output infidelity from independent logical-state errors is O(ε_l^2) with perfect distillation.Source: §III · p. 3
  • Repeated rounds require independent first-round blocks and can become unwieldy: r=3 and k=10 uses about 1.87×10^7 inputs for 10^6 outputs.Distilled outputs have correlated errors and must not meet again in a subsequent distillation circuit.Source: §III · pp. 3–4
  • A first round without correlated logical errors, such as Bravyi-Kitaev 15-to-1, avoids needing multiple blocks in the second round and can yield smaller multiround protocols.The paper reports that numerically optimal protocols take this approach.Source: §III · p. 4
  • As k→∞ and ε→0, r-round protocols consume 2^r+1 input states per output and achieve O(ε_in^(2^r)) infidelity, with scaling exponent approaching the conjectured bound.The paper reports this as the closest any protocol has come to reaching that bound.Source: §III · p. 4

IV. ANALYSIS

Numerical searches find multilevel distillation more efficient than previously studied protocols below 10^-7 output infidelity, with a fitted input-cost trend of 14 log10(1/ϵout)−40. The analysis derives error suppression for concatenated H-code distillers under ideal-operation and independent-input-error assumptions.

IV. ANALYSIS

  • The analysis assumes perfect circuit operations apart from faulty initial magic states, and independent error probabilities for each input state.The perfect-operation approximation is described as valid when fault-tolerant logical gate error is far below the final distilled infidelity.Source: §IV · p. 4
  • When both input infidelities equal ϵ, the one-level distiller's output error is (3k+1)ϵ^2, with second-order suppression from detecting single errors.The output error is conditioned on successful distillation; higher-order terms are omitted.Source: §IV · p. 4
  • The two-level distiller detects errors with probability k^2ϵl + 2(k+4)^2ϵp + 2k^2(k+4)^2ϵlϵp + (...), and similar suppression extends to higher levels.The numerical results support neglecting higher-order terms in optimally efficient protocols.Source: §IV · p. 4
  • Below 10^-7 output infidelity, multilevel distillation dominates previously studied protocols in the stated quantum-computing regime.Figure 3 plots average input states consumed against output infidelity; its dashed line is a fit of 14 log10(1/ϵout)−40.Source: §IV · pp. 4–5, Figure 3 · p. 5

V. CONCLUSIONS

The conclusion identifies H codes as useful for efficient magic-state distillation and notes possible extensions beyond Hadamard states. It also identifies total fault-tolerant circuit costs as an open area for further study.

V. CONCLUSIONS

  • H codes can distill magic states for T = exp(iπ(I − σz)/8), potentially enabling three-qubit magic-state distillation for controlled-controlled-Z, which is locally equivalent to Toffoli.The paper presents this as a possibility, with details in Appendices C and D.Source: §V · p. 5
  • The study evaluates average input-to-output efficiency, while future work is needed to examine total qubit and gate costs for fault-tolerant multilevel distillation.The conclusion points to a more rigorous assessment of these overall costs.Source: §V · p. 5
  • Multilevel distillation is described as important for large-scale fault-tolerant quantum computing, where magic-state distillation is often considered the most costly subroutine.The conclusion notes that other codes combining high density, high distance, and transversal Hadamard may be discovered; H codes are useful for their efficiency and simple construction.Source: §V · p. 5

Appendix A: Error analysis in multilevel H-code distillation

The appendix derives output-error contributions by tracing which encoded-state and physical-ancilla errors evade the Hadamard measurement and code-stabilizer checks. Concatenation raises the distance, but higher-level protocols require additional checks and their growing error-term coefficients constrain when they are useful.

Appendix A: Error analysis in multilevel H-code distillation

  • For two concatenated H codes, the [[n1n2, (n1−4)(n2−4), 4]] code has distance 4; the distance grows as 2^r for r levels.Concatenated codes use the first-level logical qubits as physical qubits for distinct second-level codes.Source: Appendix A · p. 6
  • The error checks are a Hadamard-basis measurement and code-stabilizer measurements, which detect odd parity and nonlogical error configurations, respectively.The Hadamard measurement can miss an even number of encoded-state errors; stabilizer checks detect configurations that are not logical operators.Source: Appendix A · p. 6
  • At three or more levels, repeating the controlled-Hadamard measurement makes an encoded-state error require at least 2(t−1) physical-state errors to escape detection, giving O(ϵlϵp^(2t−1)) error probability.The repeated measurement consumes 2(t−1) physical-level magic states.Source: Appendix A · p. 6
  • For one-level H codes with similar ϵl and ϵp below 0.01, the leading output-error terms are (k−1)ϵl^2 and (2k+2)ϵp^2; relevant higher-order terms can make distillation add errors.The approximation is appropriate when k is not too large.Source: Appendix A · p. 7
  • The two-level output error includes (k^2−1)ϵl^2, 8(k^2+4k+3)ϵp^4, and (k+4)^2ϵlϵp^2; the first optimal example uses k=8, ϵl=3.5×10^−5, and ϵp=9×10^−4.The example's two input types come from earlier Bravyi-Kitaev and Meier-Eastin-Knill rounds.Source: Appendix A · p. 7
  • For r≥3, error-term coefficients grow rapidly, and the search did not favor three-level protocols until output error fell below 10^−25 or four-level protocols above 10^−40.Circuit size is also identified as a possible limit on usefulness.Source: Appendix A · p. 7

Appendix B: Optimal multi-round distillation

The appendix numerically searches multi-round distillation sequences, accounting for rejection through average resource cost. The results show where multilevel protocols are most efficient and which concatenation levels are favored at very low target infidelities.

Appendix B: Optimal multi-round distillation

  • Protocols are optimal when no other sequence uses fewer average input states for a given ϵout, or achieves lower ϵout for the same average input cost.The average cost includes failure-and-repeat steps after detected errors.Source: Appendix B · p. 7
  • The search constrained rounds to r≤5, encoded logical qubits to k≤20 per H code, and multilevel codes to square layouts.The plotted protocols show only the final round; earlier rounds can use different protocols.Source: Appendix B · p. 7
  • For ϵout below 10^−7, multilevel protocols are most efficient; level-three protocols lead below 10^−25, while higher levels are not efficient above 10^−40.Higher concatenation levels up to five were included in the search.Source: Appendix B · p. 7
  • The comparison includes prior Bravyi-Kitaev, Meier-Eastin-Knill, and Bravyi-Haah protocols, and searched sequences may use any of them where efficient.The methods are cumulative, so newer schemes can also use earlier protocols.Source: Appendix B · p. 8
  • For ϵ≤0.01, the numerical error-function analysis uses the leading Taylor-series term for each protocol.The appendix identifies Bravyi-Kitaev, Meier-Eastin-Knill, and Bravyi-Haah error functions as inputs to the search.Source: Appendix B · p. 8

Appendix C: Distilling magic states for T gates with H codes

H codes can distill the magic state used to implement T gates as well as Hadamard magic states. The protocol encodes |A⟩ states, measures a controlled operator using physical-level |A⟩ states, and performs error detection.

Appendix C: Distilling magic states for T gates with H codes

  • H codes also distill |A⟩=(1/2)(|0⟩+e^(iπ/4)|1⟩), the magic state used to make the T=exp(iπ(I−σz)/8) gate.The state is stabilized by TXT†=(1/2)(X+Y), which is transversal in H codes.Source: Appendix C · p. 8
  • The construction can also be used to make Toffoli magic states.Source: Appendix C · p. 8
  • The protocol encodes |A⟩ states as logical qubits, measures controlled-(TXT†) using physical-level |A⟩ states, then performs routine error detection.Source: Appendix C · p. 8

Appendix D: Distilling Toffoli magic states

A CSS code with transversal CNOT and T operations also supports transversal CCZ, which is locally equivalent to Toffoli. The appendix presents this gate relationship as a route to distilling Toffoli magic states, assuming Clifford operations are freely available.

Appendix D: Distilling Toffoli magic states

  • A CSS+T code has transversal CCZ because CCZ decomposes into CNOT and T (or T†) gates.Source: Appendix D · p. 8, Figure 4 · p. 8
  • CCZ is locally equivalent to Toffoli, and CSS+T codes could distill three-qubit magic states for the Toffoli gate.The proposed use assumes Clifford operations are freely available.Source: Figure 4 · p. 8, Appendix D · p. 8
  • Table I specifies optimal distillation protocols using error functions to show when a round's inputs come from earlier-round outputs.The caption states that CBH and CMEK data come from Ref. [12].Source: Table I · p. 10
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