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Magic State Distillation: Not as Costly as You Think

Daniel Litinski

arXiv:1905.06903v3quant-ph

TL;DR

Magic-state distillation is widely viewed as a dominant cost in fault-tolerant quantum computing. This work tunes code distances across distillation qubits, reducing space-time cost by approximately 90% compared with the previous state of the art.

  • Problem

    The daunting resource requirements of fault-tolerant surface-code computing leave the contribution of magic-state distillation to overall cost needing clarification.

  • Method

    The authors tune the code distances of different distillation qubits to reduce distillation resource requirements.

  • Results

    Approximately 90% lower space-time cost than the previous state of the art was achieved for the constructed magic-state distillation protocols.

  • Takeaways & Limitations

    The results suggest that magic-state distillation need not be the main source of resource demands in fault-tolerant surface-code quantum computing.

  • Takeaways & Limitations

    The reported resource estimates are not rigorous simulations and depend strongly on hardware-specific error parameters and decoding procedures.

Abstract

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Despite significant overhead reductions since its first proposal, magic state distillation is often considered to be a very costly procedure that dominates the resource cost of fault-tolerant quantum computers. The goal of this work is to demonstrate that this is not true. By writing distillation circuits in a form that separates qubits that are capable of error detection from those that are not, most logical qubits used for distillation can be encoded at a very low code distance. This significantly reduces the space-time cost of distillation, as well as the number of qubits. In extreme cases, it can cost less to distill a magic state than to perform a logical Clifford gate on full-distance logical qubits.

1 Distillation circuits

Distillation circuits derive magic states from nontrivial rotation sequences equivalent to the identity, while measurements on selected qubits detect errors and enable postselection. The 15-to-1 protocol suppresses incoherent errors cubically, whereas coherent over-rotations perform substantially worse; multi-output circuits extend the approach to 20-to-4 distillation.

  • 15-to-1 distillation: Nontrivial π/8 rotation sequences equivalent to the identity can be modified to prepare a magic state alongside |+⟩ ancillas.In the 15-rotation construction, qubits 2–5 return to |+⟩ when error-free and their X measurements provide error detection; any −1 outcome causes rejection.
  • 15-to-1 distillation: 3.5 × 10^-11 is the approximate output error 35p3 at p = 10^-4, compared with the exact pout = 3.501 × 10^-11.The 15-to-1 circuit detects any single faulty rotation and no combination of two faulty gates can go undetected.
  • Error models: 10.3704p3 is the leading-order output error pout for random Pauli errors, rather than 35p3 for Z-type errors.The reduction arises because X, Y, and Z preparation faults induce P−π/4, P3π/8, or P5π/8 rotations with differing probabilities of causing an undetected error.
  • Error models: 1.22 × 10^-9 is the output magic-state infidelity for coherent over-rotation with ϕ = arcsin(1/100), nearly two orders of magnitude above the incoherent case.Although the underlying gate fidelity matches a Z error probability of 10^-4, coherent errors are detected less effectively.
  • 20-to-4 distillation: 22p2 is the leading-order output error probability for the 20-to-4 protocol under a Z-Pauli error model, which produces four magic states from 20 rotations.The circuit is obtained by modifying a 24-rotation identity circuit, and 22 rotation pairs can lead to an output error.

2 Faulty logical T gates

The section develops asymmetric surface-code layouts for faulty logical T gates, exploiting the fact that some errors in the 15-to-1 protocol are detectable. It compares state injection with faulty T measurements and gives a pessimistic ballpark error estimate for the resulting protocols.

  • Asymmetric encoding: Z errors on qubits 2–5 are detectable, so their logical Z operators can use distance dZ ≤ dX while retaining X distance dX.Their X-error probability per code cycle is 0.5(dZ/dX)·pL(pphys, dX).
  • Asymmetric encoding: The resulting arrangement uses three code distances: spatial distances dX and dZ, plus temporal distance dm.It contains one dX × dX patch, four dZ × dX patches, and two measurement ancilla regions with width dX.
  • Faulty T-gate methods: Faulty logical T gates can be implemented either through traditional state injection or through faulty T measurements.Faulty T measurements reverse the order of entanglement and the faulty T gate to avoid the Clifford correction required by state injection.
  • Faulty T-gate methods: State injection prepares a faulty magic state and requires a Pπ/4 correction with 50% probability, increasing the distillation space-time cost.The correction can require extra time or extra space.
  • Error estimation: The error estimate includes storage errors and Pπ/2, Pπ/4, and P−π/4 rotation errors, with qubit-dependent distances dH = dX for qubit 1 and dH = dZ otherwise.Storage-error probabilities include 0.5(dX/dH) · pL(pphys, dH) and 0.5(dH/dX)·pL(pphys, dX).
  • Error estimation: The error estimate is a pessimistic ballpark rather than a rigorous simulation because it does not use an actual decoder.It also includes an output-qubit Z-error probability of 0.5(l/dX) · pL(pphys, dX) · dm and Pπ/2, P−π/4, and Pπ/4 errors at pphys/3.

3 15-to-1 distillation

The 15-to-1 protocol achieves low output errors with low-distance encoding, yielding modest space-time costs. Its output fidelity is ultimately limited, motivating two-level distillation for higher-fidelity magic states.

  • Resource costs: The total 15-to-1 protocol cost is 2 · (dX + 4dZ) · 3dX + 4dm physical qubits and 6dm/(1 − pfail) code cycles.The space-cost expression includes physical measurement ancillas, while pfail is the protocol failure probability.
  • Resource costs: For pphys = 10−4, (15-to-1)7,3,3 achieves pout = 4.4 × 10−8 with a space cost of 810 qubits and a time cost of 18.1 code cycles.The protocol’s space-time cost is obtained by multiplying its qubit and cycle costs.
  • Storage-error requirements: For pout = 4.4×10−8, the required full-computation storage distance is d = 11 for 100 qubits and d = 13 for 10,000 qubits.The distance is selected as the smallest odd integer satisfying the storage-error condition.
  • Resource costs: The reported space-time costs are 5.49d3 and 3.33d3, compared with 3d3 for explicitly performing a logical CNOT on two distance-d qubits.These costs are measured in physical data qubits × code cycles, or qubitcycles.
  • Output error rates: At pphys = 10−4, (15-to-1)9,3,3 and (15-to-1)11,5,5 produce pout = 9.3 × 10−10 and pout = 1.9 × 10−11, respectively.At pphys = 10−3, (15-to-1)17,7,7 produces pout = 4.5 × 10−8.
  • Limitations: The 15-to-1 protocols cannot generate arbitrarily good output states because pout is limited by ∼10p3phys, so higher fidelity requires two-level protocols.The paper turns to two-level protocols to overcome this output-error limitation.

4 Two-level protocols

Two-level distillation uses low-distance level-1 factories to feed a second distillation round, with variants balancing error suppression, throughput, and space-time cost. The protocols achieve very low output errors at substantial but quantified qubit-cycle costs, while coherent errors remain a limitation.

  • (15-to-1) × (15-to-1): Two-level protocols feed distilled level-1 magic states into a second distillation circuit, using nL1 level-1 blocks and separate code distances for each level.The central level-2 region is fed by two level-1 regions, each containing nL1/2 blocks, with nL1 even.
  • (15-to-1) × (15-to-1): For pphys = 10−4, (15-to-1)4 9,3,3 × (15-to-1)25,9,9 produces pout = 6.3×10−25 for 1,260,000 qubitcycles.For pphys = 10−3, (15-to-1)6 11,5,5 × (15-to-1)29,11,13 produces pout = 3.3×10−14 for 3,810,000 qubitcycles.
  • 20-to-4 distillation: The (15-to-1)×(20-to-4) protocol offers cheaper, weaker suppression for target errors between one-level 15-to-1 and two-level 15-to-1 performance.Its level-2 region is longer because the 20-to-4 circuit acts on seven qubits, including four output states.
  • 20-to-4 distillation: The 9,3,3 × (20-to-4)15,7,9 protocol generates states with pout = 2.4 × 10−15.Other protocols, 13,5,5 × (20-to-4)27,13,15, achieve output errors of 1.4 × 10−10 and 2.6 × 10−11 per state at 1,410,000 and 1,840,000 qubitcycles per output state.
  • Coherent errors: Coherent errors can substantially worsen performance: the level-1 block of (15-to-1)4 13,5,5 × (20-to-4)27,13,15 outputs pout ≈10−6 versus ≈10−8 for one-level 15-to-1 at pphys = 10−3.The paper notes that a more careful treatment of coherent errors is necessary but beyond its scope.

5 Synthillation

Synthillation generates resource states for entire layers of commuting π/8 rotations rather than individual T gates, with |CCZ⟩ as the simplest example. The section shows that such protocols can reduce distillation cost, while warning that sequential output consumption may congest the distillation block.

  • Concept and construction: Synthillation produces resource states for layers of commuting π/8 rotations, including |CCZ⟩, which enables a CCZ gate represented by seven such rotations.The |CCZ⟩ state is prepared by applying CCZ to |+⟩⊗3.
  • Concept and construction: Synthillation circuits derive from ordinary distillation circuits by canceling seven rotations from a 15-rotation identity representation, yielding the seven-rotation CCZ decomposition.The construction multiplies the circuit by corresponding +π/8 and −π/8 rotations.
  • Resource costs: 2,820,000 qubitcycles produce |CCZ⟩ states with pout = 5.2 × 10^-11, whereas four ordinary magic states at comparable gate error cost 7,360,000 qubitcycles.The comparison uses four T-gate states, each with pout = 2.6 × 10^-11, because CCZ execution requires four magic states.
  • Resource costs and limitations: 447,000 qubitcycles produce output states with pout = 7.2 × 10^-14, but sequential consumption of three outputs can congest the distillation block.The congestion problem becomes more severe as generated output states must be consumed one after another.

6 Small-footprint protocols

Small-footprint distillation protocols trade increased space-time cost for substantially fewer physical qubits while achieving target output errors near 10^-9. A one-level protocol uses 762 qubits, and a two-level protocol uses 7,780 qubits but costs 3,650,000 qubitcycles per output state.

  • 15-to-1: Using one ancilla region reduces the 15-to-1 footprint to 4(dX + 4dZ)dX + 2dm physical qubits, while doubling its time cost to 12dm.The error estimate remains identical to the ordinary 15-to-1 protocol.
  • 15-to-1: 762 qubits yields pout = 1.5×10−9 in the small-footprint (15-to-1)9,3,3 protocol.Its 36.2-cycle duration gives a space-time cost of 27,600 qubitcycles, higher than comparable protocols with similar output error.
  • Two-level 15-to-1 distillation: 7,780 physical qubits yields pout = 6.1 × 10−10 in the small-footprint (15-to-1)9,5,5 × (15-to-1)21,9,11 protocol.The protocol takes 469 cycles, corresponding to 3,650,000 qubitcycles per output state.
  • Two-level 15-to-1 distillation: The two-level protocol sacrifices space-time cost: an ordinary (15-to-1) × (20-to-4) protocol reaches pout = 1.4 × 10−10 for 1,420,000 qubitcycles.This comparison illustrates the tradeoff between minimizing qubit footprint and minimizing space-time overhead.

7 Conclusion

The protocols reduce magic-state distillation space-time cost by approximately 90% versus prior work, but the estimates are based on error analysis and serve primarily as a proof of principle. The conclusion identifies substantial optimization opportunities and argues that surface-code quantum-computing costs are driven mainly by low topological-code encoding rates rather than distillation overhead.

  • 7 Conclusion: Approximately 90% lower space-time cost than the previous state of the art demonstrates that distillation overhead need not dominate fault-tolerant quantum-computing resources.The reported figures come from careful error analysis rather than full surface-code simulation with an actual decoder, so they should be treated cautiously as a proof of principle.
  • 7 Conclusion: The study leaves room for further reductions because it considers only 15-to-1, 20-to-4, and |CCZ⟩ synthillation protocols.More sophisticated distillation circuits may reduce cost further, while the broader (3k + 8)-to-k family does not obviously improve it at higher k.
  • 7 Conclusion: The space-time cost per output state has minima at k = 2 and k = 4 for even integer k in the considered protocol family.The governing expression is (3k+8)(k+3)/k, and arbitrarily many related protocols can be generated from triorthogonal codes.
  • 7 Conclusion: Low encoding rates of topological codes, requiring thousands of physical qubits per logical qubit, may contribute more to daunting resource requirements than magic-state distillation.The conclusion also notes that the best protocol combination for two-level distillation and the benefits of protocols reducing the remaining cost are unclear.

A Faulty T measurements

Pre-selection suppresses dominant low-weight errors in faulty T measurements, with only modest overhead in a representative case. Higher faulty-measurement error rates can substantially worsen achievable output errors and distillation costs, although some protocols are only mildly affected.

  • Reducing faulty T-measurement errors: Pre-selection delays measurements until nearby checks report no syndrome for two consecutive code cycles, suppressing dominant low-weight errors around the sensitive qubit.These errors arise from single-qubit faults near the blue qubit and two-qubit-gate failures during nearby syndrome readout.
  • Reducing faulty T-measurement errors: ∼1% overall time-cost increase results when pre-selection fails with 50% probability every two code cycles and d_m = 5.The patch-failure probability after 2d_m code cycles is approximately 1% in this example.
  • Impact of higher faulty-measurement error rates: 6.6 × 10^-15 replaces 2.4 × 10^-15 for the output error of the cited protocol, while its space-time cost increases by only 0.8%.This protocol remains comparatively far from the lowest achievable output error of the corresponding (15-to-1) × (20-to-4) protocol at p_phys = 10^-4.
  • Impact of higher faulty-measurement error rates: 37.5% higher space-time cost is required at p_phys = 10^-3 and p_out = 2.5 × 10^-11 when (15-to-1) × (20-to-4) is replaced by (15-to-1) × (15-to-1).At this physical error rate, one-level protocols can no longer reach p_out ≈ 10^-8.
  • Impact of higher faulty-measurement error rates: 3.5% space-time cost increase accompanies the cited protocol’s output-error change from 2.7 × 10^-12 to 6.4 × 10^-12 at p_phys = 10^-3.The lowest achievable output error for (15-to-1) × (15-to-1) protocols rises to p_out ≈ 10^-14.
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