Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1209.2426 · Physical Review A

Magic-state distillation with low overhead

Sergey B. Bravyi, Jeongwan Haah

Published Nov 27, 2012• 344 citations
arXivHTMLPDF

Abstract

We propose a family of error-detecting stabilizer codes with an encoding rate of $1/3$ that permit a transversal implementation of the gate $T=\mathrm{exp}(\ensuremath{-}i\ensuremath{\pi}Z/8)$ on all logical qubits. These codes are used to construct protocols for distilling high-quality ``magic'' states $T\left|+\right\ensuremath{\rangle}$ by Clifford group gates and Pauli measurements. The distillation overhead scales as $O({\mathrm{log}}^{\ensuremath{\gamma}}(1/\ensuremath{\epsilon}))$, where $\ensuremath{\epsilon}$ is the output accuracy and $\ensuremath{\gamma}={\mathrm{log}}_{2}(3)\ensuremath{\approx}1.6$. To construct the desired family of codes, we introduce the notion of a triorthogonal matrix, a binary matrix in which any pair and any triple of rows have even overlap. Any triorthogonal matrix gives rise to a stabilizer code with a transversal $T$ gate on all logical qubits, possibly augmented by Clifford gates. A powerful numerical method for generating triorthogonal matrices is proposed. Our techniques lead to a twofold overhead reduction for distilling magic states with accuracy $\ensuremath{\epsilon}\ensuremath{\sim}{10}^{\ensuremath{-}12}$ compared with previously known protocols.

DOI
10.1103/physreva.86.052329

Related papers

Constant-Overhead Magic State Distillation

Adam Wills, Min-Hsiu Hsieh, Hayata Yamasaki

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 $ε$, typically grows as $\mathcal{O}(\log^γ(1/ε))$. Achieving smaller overheads, i.e., smaller exponents $γ$, is highly desirable; however, all existing protocols require polylogarithmically growing overheads with some $γ> 0$, and identifying the smallest achievable exponent $γ$ 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 $\mathcal{O}(1)$ overhead, meaning the optimal exponent $γ= 0$; this improves over the previous best of $γ\approx 0.678$ due to Hastings and Haah. In our construction, we employ algebraic geometry codes to explicitly present asymptotically good quantum codes for $2^{10}$-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 $2^{10}$-dimensional qudit as a set of $10$ qubits, using stabiliser operations on qubits. The $10$-qubit magic states distilled with these codes can be converted to and from conventional magic states for the controlled-controlled-$Z$ ($CCZ$) and $T$ 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.

arXiv:2408.07764•Aug 14, 2024•4 citations

Magic State Distillation using Asymptotically Good Codes on Qudits

Michael J. Cervia, Henry Lamm, Diyi Liu +2 more

Qudits offer the potential for low-overhead magic state distillation, although previous results for asymptotically good codes have required qudit dimension $q\gg 100$ or code length $\mathcal{N}\gg 100$. These parameters far exceed experimental demonstrations of qudit platforms, and thus motivate the search for better codes. Using a novel lifting procedure, we construct the first family of good triorthogonal codes on the $\mathbb{F}_{2^{2m}}$ alphabet with $m \geq 3$ that lies above the Tsfasman-Vladut-Zink bound. These codes yield a family of asymptotically good quantum codes with transversal CCZ gates, enabling constant space overhead magic state distillation with qudit dimension as small as $q=64$. Further, we identify a promising code with parameters $[[42,14,6]]_{64}$. Finally, we show that a distilled $|{CCZ}\rangle_{2^{2m}}$ can be reduced to a $|{CCZ}\rangle_{2^n}$ state for arbitrary $n$ with a constant-depth Clifford circuit of at most 9 computational basis measurements, 12 single-qudit and 9 two-qudit Clifford gates.

quant-pharXiv:2512.21874•Dec 29, 2025•0 citations

Distillation with Sublogarithmic Overhead

Matthew B. Hastings, Jeongwan Haah

It has been conjectured that, for any distillation protocol for magic states for the $T$ gate, the number of noisy input magic states required per output magic state at output error rate $\ensuremath{\epsilon}$ is $\mathrm{\ensuremath{\Omega}}[\mathrm{log}(1/\ensuremath{\epsilon})]$. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters $⟦\ensuremath{\sum}_{i=w+1}^{m}(\genfrac{}{}{0}{}{m}{i}),\ensuremath{\sum}_{i=0}^{w}(\genfrac{}{}{0}{}{m}{i}),\ensuremath{\sum}_{i=w+1}^{r+1}(\genfrac{}{}{0}{}{r+1}{i})⟧$ for any integers $m>2r$, $r>w\ensuremath{\ge}0$, by puncturing quantum Reed-Muller codes. When $m>\ensuremath{\nu}r$, our code admits a transversal logical gate at the $\ensuremath{\nu}$th level of Clifford hierarchy. In a distillation protocol for magic states at the level $\ensuremath{\nu}=3$ ($T$ gate), the ratio of input to output magic states is $O\mathbf{(}{\mathrm{log}}^{\ensuremath{\gamma}}(1/\ensuremath{\epsilon})\mathbf{)}$, where $\ensuremath{\gamma}=\mathrm{log}(n/k)/\mathrm{log}(d)<0.678$ for some $m$, $r$, $w$. The smallest code in our family for which $\ensuremath{\gamma}<1$ is on $\ensuremath{\approx}{2}^{58}$ qubits.

arXiv:1709.03543•Physical Review Letters•Jan 31, 2018•60 citations

Zero-level $CCZ$ Distillation

Tomohiro Itogawa, Yutaka Hirano, Yutaro Akahoshi +1 more

Magic state distillation is a key component of fault-tolerant quantum computation, as it enables the implementation of non-Clifford gates such as the $T$ gate and the $CCZ$ gate via gate teleportation. However, conventional distillation protocols require a large number of logical qubits and introduce substantial spatial and temporal overhead, posing a significant bottleneck for scalable fault-tolerant quantum computation. In this work, we propose a zero-level distillation protocol that efficiently generates a high-fidelity logical $CCZ$ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal $T/T^\dagger$ operation of the $[[ 8,3,2 ]]$ code to fault-tolerantly encode the state $\overline{CCZ}|+++\rangle$, which is subsequently teleported to three surface-code logical qubits via lattice surgery. To enable teleportation between codes with different distances, we introduce adaptively initialized teleportation (AIT), a tailored initialization procedure for the surface code. Numerical simulations demonstrate that the logical error rate scales as $p_L \simeq 300 \times p^2$ with respect to the physical error rate $p$. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at $p = 10^{-3}$ and $p = 10^{-4}$, respectively, compared to conventional seven-$T$-gate approaches. The distillation circuit requires only 22 physical qubits, 3 logical qubits, and a circuit depth of 24, reducing the space-time overhead by a factor of approximately 5-10 compared to previous methods. This result highlights the practicality of $CCZ$-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional $T$-state generation.

quant-pharXiv:2605.21867•May 22, 2026•0 citations

Magic state distillation with low space overhead and optimal asymptotic input count

Jeongwan Haah, Matthew B. Hastings, David Poulin +1 more

We present an infinite family of protocols to distill magic states for T -gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the T -gate depth of the circuit scale linearly in the logarithm of the target error δ (up to log ⁡ log ⁡ 1 / δ ). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the "inner code". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the "outer code". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.

arXiv:1703.07847•Quantum•Oct 3, 2017•71 citations

Constant sized support state distillation with qubit recycling

Victor Barizien

In order to circumvent that no universal set of gates can be achieved on a given quantum error-correcting codes, additional gates can be performed through magic state injection. This however requires to have access to high fidelity magic state, which can be obtained by the distillation of low quality states to fewer higher quality ones. In this paper, we provide a family of distillation protocols able to produce diagonal states at every level of the Clifford hierarchy. This family is obtained recursively using code doubling techniques and leveraging qubit recycling, the fact that some idle qubits of the protocols can be measured and re-used. At a given level of the Clifford hierarchy, we show that the protocols we derive can be performed at arbitrary high distance on a fix number of logical qubits. This family recovers many known efficient protocols, and uncovers new ones, such as a $111 \to 1$ protocol for $|T\rangle$ state distillation at distance $7$, which are the most compact in term of volume. Finally, we discuss possible extension of this framework for distillation protocols producing more than one output state.

quant-pharXiv:2609.17044•Sep 16, 2026•0 citations
We propose a family of error-detecting stabilizer codes with an encoding rate of 1/31/31/3 that permit a transversal implementation of the gate T=exp(\ensuremath−i\ensuremathπZ/8)T=\mathrm{exp}(\ensuremath{-}i\ensuremath{\pi}Z/8)T=exp(\ensuremath−i\ensuremathπZ/8) on all logical qubits. These codes are used to construct protocols for distilling high-quality ``magic'' states T\left|+\right\ensuremath{\rangle} by Clifford group gates and Pauli measurements. The distillation overhead scales as O(log\ensuremathγ(1/\ensuremathϵ))O({\mathrm{log}}^{\ensuremath{\gamma}}(1/\ensuremath{\epsilon}))O(log\ensuremathγ(1/\ensuremathϵ)), where \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ is the output accuracy and \ensuremathγ=log2(3)\ensuremath≈1.6\ensuremath{\gamma}={\mathrm{log}}_{2}(3)\ensuremath{\approx}1.6\ensuremathγ=log2​(3)\ensuremath≈1.6. To construct the desired family of codes, we introduce the notion of a triorthogonal matrix, a binary matrix in which any pair and any triple of rows have even overlap. Any triorthogonal matrix gives rise to a stabilizer code with a transversal TTT gate on all logical qubits, possibly augmented by Clifford gates. A powerful numerical method for generating triorthogonal matrices is proposed. Our techniques lead to a twofold overhead reduction for distilling magic states with accuracy \ensuremathϵ\ensuremath∼10\ensuremath−12\ensuremath{\epsilon}\ensuremath{\sim}{10}^{\ensuremath{-}12}\ensuremathϵ\ensuremath∼10\ensuremath−12 compared with previously known protocols.
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.
Qudits offer the potential for low-overhead magic state distillation, although previous results for asymptotically good codes have required qudit dimension q≫100q\gg 100q≫100 or code length N≫100\mathcal{N}\gg 100N≫100. These parameters far exceed experimental demonstrations of qudit platforms, and thus motivate the search for better codes. Using a novel lifting procedure, we construct the first family of good triorthogonal codes on the F22m\mathbb{F}_{2^{2m}}F22m​ alphabet with m≥3m \geq 3m≥3 that lies above the Tsfasman-Vladut-Zink bound. These codes yield a family of asymptotically good quantum codes with transversal CCZ gates, enabling constant space overhead magic state distillation with qudit dimension as small as q=64q=64q=64. Further, we identify a promising code with parameters [[42,14,6]]64[[42,14,6]]_{64}[[42,14,6]]64​. Finally, we show that a distilled ∣CCZ⟩22m|{CCZ}\rangle_{2^{2m}}∣CCZ⟩22m​ can be reduced to a ∣CCZ⟩2n|{CCZ}\rangle_{2^n}∣CCZ⟩2n​ state for arbitrary nnn with a constant-depth Clifford circuit of at most 9 computational basis measurements, 12 single-qudit and 9 two-qudit Clifford gates.
It has been conjectured that, for any distillation protocol for magic states for the TTT gate, the number of noisy input magic states required per output magic state at output error rate $\ensuremath{\epsilon}$ is $\mathrm{\ensuremath{\Omega}}[\mathrm{log}(1/\ensuremath{\epsilon})]$. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters $⟦\ensuremath{\sum}_{i=w+1}^{m}(\genfrac{}{}{0}{}{m}{i}),\ensuremath{\sum}_{i=0}^{w}(\genfrac{}{}{0}{}{m}{i}),\ensuremath{\sum}_{i=w+1}^{r+1}(\genfrac{}{}{0}{}{r+1}{i})⟧$ for any integers m>2rm>2rm>2r, $r>w\ensuremath{\ge}0$, by puncturing quantum Reed-Muller codes. When $m>\ensuremath{\nu}r$, our code admits a transversal logical gate at the $\ensuremath{\nu}$th level of Clifford hierarchy. In a distillation protocol for magic states at the level $\ensuremath{\nu}=3$ (TTT gate), the ratio of input to output magic states is $O\mathbf{(}{\mathrm{log}}^{\ensuremath{\gamma}}(1/\ensuremath{\epsilon})\mathbf{)}$, where $\ensuremath{\gamma}=\mathrm{log}(n/k)/\mathrm{log}(d)<0.678$ for some mmm, rrr, www. The smallest code in our family for which $\ensuremath{\gamma}<1$ is on $\ensuremath{\approx}{2}^{58}$ qubits.
Zero-level CCZCCZCCZ Distillation
Magic state distillation is a key component of fault-tolerant quantum computation, as it enables the implementation of non-Clifford gates such as the TTT gate and the CCZCCZCCZ gate via gate teleportation. However, conventional distillation protocols require a large number of logical qubits and introduce substantial spatial and temporal overhead, posing a significant bottleneck for scalable fault-tolerant quantum computation. In this work, we propose a zero-level distillation protocol that efficiently generates a high-fidelity logical CCZCCZCCZ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal T/T†T/T^\daggerT/T† operation of the [[8,3,2]][[ 8,3,2 ]][[8,3,2]] code to fault-tolerantly encode the state CCZ‾∣+++⟩\overline{CCZ}|+++\rangleCCZ∣+++⟩, which is subsequently teleported to three surface-code logical qubits via lattice surgery. To enable teleportation between codes with different distances, we introduce adaptively initialized teleportation (AIT), a tailored initialization procedure for the surface code. Numerical simulations demonstrate that the logical error rate scales as pL≃300×p2p_L \simeq 300 \times p^2pL​≃300×p2 with respect to the physical error rate ppp. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at p=10−3p = 10^{-3}p=10−3 and p=10−4p = 10^{-4}p=10−4, respectively, compared to conventional seven-TTT-gate approaches. The distillation circuit requires only 22 physical qubits, 3 logical qubits, and a circuit depth of 24, reducing the space-time overhead by a factor of approximately 5-10 compared to previous methods. This result highlights the practicality of CCZCCZCCZ-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional TTT-state generation.
We present an infinite family of protocols to distill magic states for T -gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the T -gate depth of the circuit scale linearly in the logarithm of the target error δ (up to log ⁡ log ⁡ 1 / δ ). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the "inner code". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the "outer code". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.
In order to circumvent that no universal set of gates can be achieved on a given quantum error-correcting codes, additional gates can be performed through magic state injection. This however requires to have access to high fidelity magic state, which can be obtained by the distillation of low quality states to fewer higher quality ones. In this paper, we provide a family of distillation protocols able to produce diagonal states at every level of the Clifford hierarchy. This family is obtained recursively using code doubling techniques and leveraging qubit recycling, the fact that some idle qubits of the protocols can be measured and re-used. At a given level of the Clifford hierarchy, we show that the protocols we derive can be performed at arbitrary high distance on a fix number of logical qubits. This family recovers many known efficient protocols, and uncovers new ones, such as a 111→1111 \to 1111→1 protocol for ∣T⟩|T\rangle∣T⟩ state distillation at distance 777, which are the most compact in term of volume. Finally, we discuss possible extension of this framework for distillation protocols producing more than one output state.