Multilevel distillation of magic states for quantum computing

Cody Jones∗

Edward L. Ginzton Laboratory, Stanford University, Stanford, California 94305-4088, USA

We develop a procedure for distilling magic states used in universal quantum computing thatrequires substantially fewer initial resources than prior schemes. Our distillation circuit is based ona family of concatenated quantum codes that possess a transversal Hadamard operation, enablingeach of these codes to distill the eigenstate of the Hadamard operator. A crucial result of thisdesign is that low-fidelity magic states can be consumed to purify other high-fidelity magic statesto even higher fidelity, which we call “multilevel distillation.” When distilling in the asymptoticregime of infidelity ε → 0 for each input magic state, the number of input magic states consumedon average to yield an output state with infidelity O(ε2

r

) approaches 2r + 1, which comes closeto saturating the conjectured bound in [Phys. Rev. A 86, 052329]. We show numerically thatthere exist multilevel protocols such that the average number of magic states consumed to distillfrom error rate εin = 0.01 to εout in the range 10−5 to 10−40 is about 14 log10(1/εout) − 40; theefficiency of multilevel distillation dominates all other reported protocols when distilling Hadamardmagic states from initial infidelity 0.01 to any final infidelity below 10−7. These methods are animportant 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

Quantum computing can potentially solve a hand-ful of otherwise intractable problems, such as factor-ing large integers [1] or simulating quantum physics [2].Though the number of applications with a known “quan-tum speed-up” is small, some are quite valuable, likethe preceding examples. Quantum computations de-pend on coherent entangled states which are very sen-sitive to noise, so fault-tolerant quantum computing ad-dresses imperfections in physical hardware with error-correcting codes [3, 4], the most studied of which arestabilizer codes [5]. However, while quantum codes pro-tect against noise, no code natively supports a univer-sal set of transversal gates for simulating any quantumcircuit [6, 7]. To achieve universal quantum computingwith error correction, Bravyi and Kitaev proposed a so-lution [8] that has received considerable attention: injectfaulty “magic states” into the code, purify them using theerror-corrected gates, then consume them to implementotherwise unavailable quantum circuits. These states are“magic” because it is possible to distill a subset of high-fidelity states from an ensemble of faulty states and be-cause they enable universal fault-tolerant quantum com-putation.

Magic state distillation has been the subject of intenseinvestigation in recent years. Knill independently intro-duced a distillation procedure for |H〉, the (+1) eigen-state of the Hadamard operation [9], prior to the work byBravyi and Kitaev [8]. Reichardt showed that these pro-tocols were equivalent and introduced an improvementwhich increased the threshold error rate [10]. More re-cently, Meier et al. introduced a 10-to-2 distillation pro-cedure based on a code with two encoded qubits [11],

∗ ncodyjones@gmail.com

and Bravyi and Haah introduced a (3k + 8)-to-k pro-cedure using so-called triorthogonal codes with even kencoded qubits [12]. The distillation procedures we de-velop herein continue this trend of using larger, multi-qubit codes. The relationship of magic-state distillationto other aspects of quantum information has also been anarea of active study. Fowler and Devitt have proposedmethods to reduce the size of distillation circuits whenusing topological quantum error correction [13]. Magic-state distillation has been demonstrated experimentallyin NMR [14]. Additionally, distillation protocols for qu-dits have been proposed and analyzed [15, 16].

For a quantum state, we quantify the probability ofit having an error using the infidelity 1 − F , whereF = 〈ψ| ρ |ψ〉 is the fidelity between some mixed stateρ and the ideal state |ψ〉. The initial |H〉 states are pre-pared in a faulty manner before being injected into afault-tolerant quantum code, and Reichardt proved thatthe theoretical-limit infidelity for |H〉 states to be dis-tillable is about 0.146 [10]. Campbell and Browne ex-amined further properties of mixed states that may bedistilled [17]. The efficiency of distilling high-infidelitymagic states to low infidelity is of great importanceto fault-tolerant quantum computing. Although magicstates are the widely preferred method for achieving uni-versality, distillation circuits are currently estimated torequire the majority of resources in a quantum com-puter [18, 19]. Therefore, advances in distillation pro-tocols are important steps toward making quantum com-puting possible.

This paper presents two important, related results.First, we specify a family of [[n, (n − 4), 2]] Calderbank-Shor-Steane (CSS) quantum stabilizer codes [20, 21]known as “H codes” with transversal Hadamard opera-tion, for even n ≥ 6. A “transversal” quantum operationis one where a gate acting on a logical, encoded qubitis implemented by independent gates on each qubit inthat code block (see p. 483 of Ref. [4]). The H codes

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are dense, because the ratio of logical qubits to physicalqubits (n−4)/n→ 1 as n→∞. Second, we demonstratethat concatenated versions of H codes allow for distilla-tion of high-fidelity encoded magic states by consuminglow-fidelity magic-state ancillas. We call this “multileveldistillation,” because each such protocol takes two typesof magic-state inputs, which have different levels of in-fidelity and which are applied at different concatenationlevels within the distillation circuit. Multilevel protocolslead to the most efficient procedure for distilling magicstates reported so far. For suitably small infidelity ε ineach input magic state with independent errors, there ex-ists a sequence of multilevel protocols that yields outputmagic states with infidelity O(ε2

r

) and requires asymp-totically 2r + 1 input states per distilled output state.This efficiency comes close to the “optimality” boundconjectured in Ref. [12]. For the purposes of developingquantum devices, this result is useful for exposing limitsfor optimizing quantum error correction. While this re-sult is interesting theoretically, we also numerically studythe distillation efficiency for εin = 0.01, which is of prac-tical importance to fault-tolerant quantum computing.We find that multilevel distillation is superior to previ-ously reported protocols when the final infidelity is below10−7.

Throughout this paper, we adopt the following no-tation for single-qubit Pauli operators, for readability:X ≡ σx, Z ≡ σz, and I is the identity operator on aqubit. Additionally, we use “physical qubit” to denotethose qubits used to produce a quantum code, whereas“logical qubits” are the protected information inside thecode, again for readability. It may be the case that phys-ical qubits for one encoding level are themselves the logi-cal qubits of another code, which is a standard techniqueof quantum code concatenation [3, 4, 22].

II. A FAMILY OF CODES WITHTRANSVERSAL HADAMARD

We define a family of CSS quantum codes which encodean even number k logical qubits using (k + 4) physicalqubits and possess a transversal Hadamard operation, sowe call them collectively “H codes” and denote Hn asthe code using n = k + 4 physical qubits. Any H codemay be defined as follows. The stabilizer generators areS1 = X1X2X3X4, S2 = Z1Z2Z3Z4, S3 = X1X2X5X6 . . . Xn,S4 = Z1Z2Z5Z6 . . . Zn, where subscripts index over physi-cal qubits and tensor product between Pauli operators isimplicit. The logical Pauli operators (corresponding tological qubits), denoted with an over bar and indexed byi = 1 . . . k, are Xi = X1X3Xi+4 and Zi = Z1Z3Zi+4. TheHadamard transform exchanges X and Z operators, so ap-plication of transversal Hadamard gates at the physicallevel enacts a transversal Hadamard operation at the log-ical level, which will be a useful property when we laterconcatenate these codes. All H codes have distance two,which means they can detect a single physical Pauli er-

HH

HH

HH

Enco

de

Dec

ode

0000HH

0 H H Hadamard-Basis

Measurement

PhysicalError

Detection

DistilledMagicStates

H R (-π/4)Y R (π/4)Y=

(b)

(a)

HH

FIG. 1. Distillation of |H〉 magic states using anH code. (a) Controlled-Hadamard gate is constructed us-ing RY (−iπ/4) = exp(iπσY /8) and its inverse, each of whichrequires one |H〉 state [11]. (b) Initial |H〉 states (left) are en-coded with four additional qubits, initialized to |0〉 here. Theboxes “Encode” and “Decode” represent quantum circuits forencoding and decoding, which are not shown here.

ror. The product of two logical Pauli operators of thesame type for two distinct logical qubits has weight two(number of non-identity physical, single-qubit Pauli op-erators); the product of same-type Pauli operators onall logical qubits is also weight-two at the physical level.The stabilizers come in matched X/Z pairs, so there areno weight-one logical operators.

The (+1) eigenstate |H〉 = cos(π/8) |0〉 + sin(π/8) |1〉of the Hadamard operator H = (1/

√2)(X + Z) is a magic

state for universal quantum computing [8–12]. In par-ticular, two of these magic states can be consumed toimplement a controlled-H operation [9, 11], enabling oneto measure in the basis of H (see Fig. 1a). Our dis-tillation procedure is as follows: (a) encode faulty |H〉magic states in an H code; (b) measure in the basis ofthe transversal Hadamard gate by consuming |H〉 ancil-las; (c) reject the output states if either the measure-Hadamard or code-stabilizer circuits detect an error. Forexample, when an H(k+4) code is used for distillation, k|H〉 states are encoded as logical qubits using (k+4) phys-ical qubits. Each transversal controlled-Hadamard gateconsumes two |H〉 states [11], and this gate is appliedto all physical qubits, which results in the (3k + 8)-to-kinput/output distillation efficiency of these codes. A di-agram of the quantum circuit for distillation using H6 isshown in Fig. 1b.

III. MULTILEVEL DISTILLATION

Multilevel distillation uses concatenated codes withtransversal Hadamard for distillation, in such a man-ner that the protocol takes as input magic states at twodifferent levels of infidelity, and the two types of magic

3

Level-1encoding

Level-2encoding

Level-1encoding

(b)

(a)

Physical qubits

Logical qubits

Physical qubits Logical qubits

FIG. 2. Concatenation of H codes. (a) Six physical qubitsare coupled into an H6 code with two logical qubits (b) A6×6 array of physical qubits are coupled into a concatenatedtwo-level H6 code.

states enter at different concatenation levels in the code.The |H〉 ancillas consumed for transveral controlled-Hadamard measurement are of lower fidelity than theencoded logical |H〉 states being distilled. When twoquantum codes with transversal Hadamard are concate-nated, the resulting code also has transversal Hadamard.Under appropriate conditions, the distance of the con-catenated code is the product of the distances for theindividual codes: d′ = d1d2 [11]. Thus the concatenationof two H codes yields a distance-4 code with transversalHadamard, and r-level concatenation has distance 2r.

The concatenation conditions for H codes are that,through all levels of concatenation, any pair of physicalqubits have at most one encoding block (at any level) incommon. The reasons for this restriction are that logi-cal errors in the same block are correlated and that thestatement above that distance multiplies through con-catenation assumes independence of errors, so two qubitsfrom the same encoding block can never be paired againin a different encoding block. The required arrangementof qubits can be given a geometric interpretation. Ar-range all physical qubits at points on a cartesian grid inthe shape of a rectangular solid, with the number of di-mensions given by the number of levels of concatenation.A square, cube, or hypercube are possible examples atdimensionality two, three, or four. Each dimension is as-sociated with a level of concatenation, and there mustbe an even n ≥ 6 qubits in each dimension to form anH code. Construct H codes in the first dimension byforming an encoding block with each line of qubits in thisdirection, as in Fig. 2a. This will give rise to k = n − 4logical qubits along each line in this direction. Repeatthis procedure by grouping these first-level logical qubitsin lines along the second dimension to form logical qubitsin a two-level concatenated code, as in Fig. 2b. Contin-uing in this fashion through all dimensions ensures thatany pair of qubits have at most one encoding block incommon.

As with the H codes, multilevel codes use a transversal

logical Hadamard-basis measurement to detect whetherany one encoded qubit has an error (an even number ofencoded errors would not be detected). If the logical |H〉states have independent error probabilities εl, then thedistilled states will have infidelity O(εl

2) with perfect dis-tillation. We must also consider whether the Hadamard-basis measurement has an error. For a two-level codearranged as a square of side length n, the transversalcontrolled-Hadamard gates at the lowest physical levelrequire (2n2) |H〉 magic states, each of which has infi-delity εp. However, this is a distance-4 code, so for in-dependent input error rates, the probability of failing todetect errors at the physical level is O(εp

4) + O(εlεp2)

(rigorous analysis is provided later). The code can de-tect more errors in the magic states at the lower physicallevel, so these |H〉 states can be of lower fidelity thanthe magic states encoded as logical qubits and success-fully perform distillation. This is the essential distinctionbetween multilevel distillation and all prior distillationprotocols. When multiple rounds of distillation are re-quired [19], low-fidelity magic states are less expensive toproduce, so multilevel protocols achieve higher efficiency.

Multilevel distillation protocols are applied in rounds,beginning with a small protocol (such as an H code) andprogressing to concatenated multilevel codes. Let us de-note the output infidelity from a single round by the func-tion εout = En1×...nt

t (εl, εp). For each such function, t isthe dimensionality (number of levels of concatenation)and n1 . . . nt are the sizes of each dimension, which neednot all be the same. As before, εl and εp refer to the inde-pendent error probabilities on logical and physical magicstates, respectively. A typical progression of rounds us-ing a source of |H〉 states with infidelity ε0 might beε1 = En1 (ε0, ε0), ε2 = En×n2 (ε1, ε0), etc.

Multilevel distillation circuits tend to be much largerin both qubits and gates than other protocols. Becausethere can be many encoded qubits, the protocol is stillvery efficient, but the size of the overall circuit may be aconcern for some quantum computing architectures. Atany number of levels, the distilled output states havecorrelated errors, so distilled magic-state qubits in ourprotocol must never meet again in a subsequent distil-lation circuit (we require that errors are independentwithin the same encoding block, as in Refs. [11, 12]).Let us suppose that one performs two rounds of dis-tillation, where the first round uses one-level distillerswith k encoded magic states and the second round usestwo-level distillers with k2 encoded states. Because theinputs to each distiller in the second round must haveindependent errors, there must be k2 independent dis-tillation blocks in the first round. Therefore, to dis-till k3 output states through two rounds, we require:k3[logical inputs] + 2k2(k + 4)[physical inputs] + 2k(k +4)2[physical inputs] = 5k3 + 24k2 + 32k input states.

Consider a similar sequence through r rounds witheach distiller in round q having kq encoded qubits. Thetotal number of logical magic states is kr×kr−1× . . . k =kr(r+1)/2 to ensure that errors are independent between

4

logical magic states in every round. In the first round, thenumber of consumed magic states is 2(k+ 4)kr(r+1)/2−1;in any subsequent round q ≥ 2, the number of con-sumed magic states is 2q−1(k+4)qkr(r+1)/2−q (recall thatthe Hadamard measurement is implemented 2q−2 times,meaning it is repeated for q ≥ 3). The total number ofinput magic states can thus be expressed as[

1 +k + 4

k+

r∑q=1

2q−1(k + 4

k

)q]kr(r+1)/2. (1)

For r = 2, this reproduces the expression above. Whatalso becomes clear is that the total size of multilevel pro-tocols becomes unwieldy as r and k increase. For ex-ample, the case of r = 3 and k = 10 would requireabout 1.87 × 107 input magic states and a comparablenumber of quantum gates to distill 106 output magicstates. However, since efficient multilevel distillation pro-tocols, measured in the ratio of low-fidelity |H〉 inputstates consumed to yield a single high-fidelity |H〉 out-put, use k � 1 and multiple rounds, the greatest bene-fit from their application is seen in large-scale quantumcomputing, where a typical algorithm run may require1012 magic states, each with error probability 10−12 [19].Moreover, alternative designs can circumvent these is-sues. If the first round uses a different protocol with-out correlated errors across logical magic states, such asBravyi-Kitaev 15-to-1 distillation, then having multipledistillation blocks is unnecessary in the second round us-ing a two-level concatenated protocol, which would leadto smaller multi-round, multilevel protocols. Indeed,Sec. IV shows that optimal protocols found by numer-ical search happen to take this approach.

The “scaling exponent” γ of a distillation protocolcharacterizes its efficiency. Specifically, O(logγ(εin/εout))input states are required to distill one magic state of infi-delity εout. Scaling exponents for previous protocols areγ ≈ 2.46 (“15-to-1” [8, 9]), γ ≈ 2.32 (“10-to-2” [11]), andγ ≈ 1.6 (triorthogonal codes [12]). Moreover, Bravyi andHaah conjecture that no magic-state distillation protocolhas γ < 1 [12]. In this work, if each round of distillationuses one higher level of concatenation in the multilevelprotocols, then the number of consumed inputs doubles.In the limits of k → ∞, ε → 0, multilevel protocols re-quire 2r+1 input states to each output state for r roundsof distillation, where the rth round is a level-r distiller.The final infidelity is O((εin)2

r

), so the scaling exponentis γ = log(2r + 1)/ log(2r) → 1 as r → ∞, which is theclosest any protocol has come to reaching the conjecturedbound. We show later through numerical simulation thatγ ≈ 1 for error rates relevant to quantum computing.

IV. ANALYSIS

We make the conventional assumption that all quan-tum circuit operations are perfect, except for the initial|H〉 magic states we intend to distill. This is a valid ap-

proximation if all operations are performed using fault-tolerant quantum error correction where the logical gateerror is far below the final infidelity for distilled magicstates [3, 19]; for a more explicit construction of fault-tolerant distillation circuits, see Ref. [13]. Additionally,following the methodology in Refs. [8, 11], we can con-sider each magic state with infidelity ε as the mixed stateρ = (1−ε) |H〉 〈H|+ε |−H〉 〈−H|, where |−H〉 is the (−1)eigenstate of the Hadamard operation.

Determining the infidelity at the output of distillationbecomes simply a matter of counting the distinct waysthat errors lead to the circuit incorrectly accepting faultystates. This process is aided by the geometric picturefrom earlier, and details are given in Appendix A. It isessential that error probabilities εl and εp for each inputmagic state are independent. Then a one-level, (3k+ 8)-to-k distiller using the H(k+4) code has output error rateon each |H〉 state as

E(k+4)1 (εl, εp) = (k − 1)εl

2 + (2k + 2)εp2 + (. . .), (2)

where higher order terms denoted (...) are omitted. Ournumerical results justify the use of lowest-order approxi-mations as higher-order terms are negligible in optimallyefficient protocols. The lowest-order error rates are bothsecond order, because the Hadamard basis measurementand H(k+4) code can together detect a single error inany magic state. The probability of the distiller detect-ing an error, in which case the output is discarded, iskεl + 2(k + 4)εp + (. . .). If εl = εp = ε, then the outputerror rate of (3k+1)ε2 conditioned on success is the sameas in Ref. [12]. Using the two-level distiller constructedfrom concatenated H(k+4) codes, the output infidelity foreach distilled |H〉 state is

E(k+4)×(k+4)2 (εl, εp) = (k2 − 1)εl

2 + 8(k2 + 4k + 3)εp4

+(k + 4)2εlεp2 + (. . .). (3)

The probability of the two-level distiller detecting an er-ror is k2εl + 2(k+ 4)2εp + 2k2(k+ 4)2εlεp + (. . .). Similarerror suppression extends to higher multilevel protocols,as examined in Appendix A.

Figure 3 shows the performance of optimal multi-rounddistillation protocols identified by numerical search, indi-cating the number of input states with ε0 = 0.01 requiredto reach a desired output infidelity εout. The markersindicate the type of protocol in the last round of distilla-tion, including Bravyi-Kitaev [8], Meier-Eastin-Knill [11],and multilevel H codes (see Appendix B for details). Thesearch attempts to identify the best distillation routinesusing any combination of known methods. Note that therecent Bravyi-Haah protocols [12] have the same perfor-mance as one-level H codes. As expected, there is atrend of using higher-distance multilevel protocols in thelast round as the output error rate εout decreases (ear-lier rounds may use different protocols). Where present,open markers indicate the best possible performance ofpreviously studied protocols without the advent of mul-tilevel distillation, and multilevel distillation is dominant

5

0 5 10 15 20 25 30 35 400

100

200

300

400

500

log10

(1/εout

)

Inpu

t mag

ic s

tate

s

One−level HTwo−level HThree−level H10−to−215−to−1

FIG. 3. (Color online) Average number of input |H〉 stateswith εin = 0.01 consumed to produce a single output |H〉 statewith fidelity εout. Multiple-round distillation can use differentprotocols in each round, and the markers indicate just thelast round of distillation. The grey-shaded squares, triangles,and circles show, respectively, the best distillation possiblewith only 15-to-1 [8], 10-to-2 [11], and triorthogonal-code [12]protocols. The dashed line is a linear fit 14 log10(1/εout)−40.

for εout ≤ 10−7, which is the regime pertinent to quan-tum computing. Moreover, in this regime, input errorrates are sufficiently small that only lowest-order termsin the E(·) output-error functions are significant. Thelinear fit provides empirical evidence that the scaling ex-

ponent is γ ≈ 1 in this regime, which demonstrates thatmultilevel protocols are close to the conjectured optimalperformance in practice.

V. CONCLUSIONS

H codes can distill magic states for T = exp(iπ(I −σz)/8), which may enable distillation of three-qubitmagic states for controlled-controlled-Z, which is locallyequivalent to the Toffoli gate [4] (see Appendices C and Dfor details). As a first pass at studying distillation proto-cols, this work considered only average input-to-outputefficiency. Future work will more rigorously examineto entire costs in qubits and gates required to fault-tolerantly distill magic states using multilevel codes [23].Multilevel distillation is an important development forlarge-scale, fault-tolerant quantum computing, where thedistillation of magic states is often considered the mostcostly subroutine [18, 19]. Other codes with high den-sity, high distance, and transversal Hadamard may yetbe discovered, though for the present, H codes are usefulfor their high efficiency and simple construction.

The author would like to thank Sergey Bravyi andBryan Eastin for assistance in analyzing protocols at lev-els three and higher. This work was supported by theUniv. of Tokyo Special Coordination Funds for Promot-ing Science and Technology, NICT, and the Japan So-ciety for the Promotion of Science (JSPS) through itsFIRST Program.

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putation: Schemes,” (2004), Preprint arXiv:quant-ph/0402171.

[10] B. W. Reichardt, Quantum Information Processing 4,251 (2005).

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[12] S. Bravyi and J. Haah, Phys. Rev. A 86, 052329 (2012).[13] A. G. Fowler and S. J. Devitt, “A bridge to lower

overhead quantum computation,” (2012), PreprintarXiv:1209.0510v3.

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6

Appendix A: Error analysis in multilevel H-codedistillation

The multilevel codes analyzed here use concatenatedH codes. When two H codes are concatenated, the logi-cal qubits of the first level of encoding are used as physicalqubits for completely distinct codes at the second level.Consider a two-level scheme: if the codes at first andsecond levels are [[n1, (n1 − 4), 2]] and [[n2, (n2 − 4), 2]],respectively, then the concatenated code is [[n1n2, (n1 −4)(n2−4), 4]], as shown in Fig. 2b of the main text. Thisprocess can be extended to higher levels of concatenation.

Determining the potential errors and their likelihoodin multilevel protocols requires careful analysis. Let usenumerate the error configurations which are detected bythe protocols; the error probability is given by summingthe probability of all error configurations that are not de-tected and that lead to error(s) in the encoded |H〉 states.As a first step, we may simplify the analysis of multilevelcodes by considering each input magic state to our quan-tum computer as having an independent probability ofσY error, as discussed in Refs. [8, 11]. This allows usto consider only one type of error stemming from eachmagic state used in the protocol.

Identifying undetected error events in multilevel distil-lation, which lead to output error rate, is aided by thegeometric picture introduced in the main text. Qubitswhich will form the code are arranged in a rectangularsolid, then grouped in lines along each dimension for en-coding. There are two error-detecting steps which to-gether implement distillation: the Hadamard-basis mea-surement and the error detection of the H codes. TheHadamard measurement registers an error for odd par-ity in the total of encoded state errors and physical-levelerrors in the first round of RY (−π/4) gates, and there isone of these for each qubit site in the code (see Fig. 1 ofthe main text).

The second method for H codes to detect errors isby measuring the code stabilizers. The code stabilizersdetect any configuration of errors which is not a logi-cal operator in the concatenated code. Because of theredundant structure using overlapping H codes, only avery small fraction of error configurations evade detec-tion. Before moving on, note that at each qubit site,there are two faulty gates applied, and two errors on thesame qubit will cancel (however, the first error will prop-agate to the Hadamard-basis measurement). Conversely,a single error in one of the two gates will propagate tothe stabilizer-measurement round, but only an error inthe first gate will also propagate to the Hadamard mea-surement. The stabilizer-measurement round will only“see” the odd/even parity of the number of errors at eachqubit site.

One type of error event that occurs at concatena-tion levels three and higher requires special treatment.If there is an error in an encoded magic state and er-rors on two physical states used for the same controlled-Hadamard gate at the physical level, then this combina-

tion of input errors is not detected by the distillationprotocol, leading to logical output error. This eventleads to the O(εlεp

2) error probability from the maintext, which is not an issue for two-level protocols, butit must be addressed in levels three and higher. Thesolution for t-level distillation, where t ≥ 3, is to re-peat the controlled-Hadamard measurement 2(t−2) times,consuming 2(t−1) magic states at the physical level. Af-ter each transversal controlled-Hadamard, the code syn-drome checks for detectable error patterns. With thisprocedure, one encoded-state error would also require atleast 2(t−1) errors in physical-level magic states to go un-detected, leading to probability of error that scales as

O(εlεp2t−1

).

Consider the pattern of errors after the two potentiallyfaulty gates on each qubit in the t-dimensional cartesiangrid arrangement. The many levels of error checking inthe H codes can detect a single error in any encodingblock at any encoding level. For this analysis, let us sep-arate the (k+4) qubits in a single H code block into twogroups: the first four qubits are “preamble” qubits, whilethe remaining k qubits are index qubits. The reason fordistinction is that the logical Yi operators, which wouldalso be undetected error configurations, have commonphysical-qubit operators in the preamble, with a degen-eracy of two: Yi = −Y1Y3Yi+4 = −Y2Y4Yi+4, because ofthe stabilizer Y1Y2Y3Y4. Conversely, the logical opera-tors are distinguished by the ith logical Pauli operatorhaving a physical Pauli operator on the ith index qubit(numbered (i+ 4) when preamble is included).

With the preamble/index distinction, we can now iden-tify the most likely error patterns. For any size H code,there are two weight-2 errors in the preamble: Y1Y2 andY3Y4. Logically, these represent the product of Y opera-tors on all encoded qubits. In the index qubits, any pairof errors is logical: Yi+4Yj+4 = YiYj . However, a pair oferrors split with one each in preamble and index is al-ways detectable by the code stabilizers. Thus, any singleencoded qubit could have a logical error stemming from apair of errors in two different configurations in the pream-ble or (k − 1) configurations in the index qubits. Thereis also one weight-three error. Each physical-state errorconfiguration is multiplied by a degeneracy factor thatis the number of ways an even number of errors occurbefore the CNOT in Fig. 1, thereby evading the Hadamardmeasurement. Thus the probability of logical error is2(k + 1)εp

2 + 4εp3 + O(εp

4). The Hadamard measure-ment fails to detect an even number of errors in the log-ical input states. There are (k − 1) ways that a pair ofencoded input errors could corrupt any given qubit and(k − 1)(k − 2)(k − 3)/6 ways four errors could corruptany given qubit (assuming k ≥ 4). This contributes errorterms (k − 1)εl

2 + (1/6)(k − 1)(k − 2)(k − 3)εl4 +O(ε6).

Finally, it is possible for a single logical error and anodd number of physical errors before the CNOT in Fig. 1of the main text, potentially in conjunction with otherphysical errors after CNOT, to occur simultaneously in away that evades both checks. This contributes a term

7

(k + 4)εlεp2 + 8(k − 1)εlεp

3 +O(εlεp4).

The numerical analysis detailed below shows that ef-ficient use of one-level H codes has similar error ratesfor εl and εp, and both are below 0.01, so the relevantterms in the error functions for one-level H codes areE

(k+4)1 (εl, εp) = (k − 1)εl

2 + (2k + 2)εp2 + (. . .), which

reproduces Eqn. (1) of the main text. As a result, thehigher-order terms above can be neglected for this rangeof parameters so long as k is not too large. Simply put, ifthe higher terms become relevant (i.e. εl, εp, or k is suffi-ciently large in magnitude), then the distillation protocolis being used ineffectively, and it may in fact cause moreerrors than it corrects. These findings are supported bythe numerical search for optimal protocols, and we pro-ceed using this approximation.

When H codes are concatenated, the analysis of un-detected error patterns becomes more complicated. Inparticular, logical errors from one layer of encoding mustbe “matched” with errors from other encoding blocksto go undetected at the next level. Consider the caseof the level-two-concatenated, square-array distiller, andfocus on one of the encoded states. As before, a pair ofencoded-state input errors evades the Hadamard mea-surement, which contributes a term (k2 − 1)εl

2. Theundetected errors resulting from consumed magic statesare more complicated. Within the upper encoding block,there are two ways a logical error could be caused by apair of errors in the preamble, and (k−1) possibilities forlogical error from a pair of index errors. However, eachof the inputs to the second level are the logical qubits ofdistinct H codes at the first level, which has additionalerror detection. The most likely errors from the first levelcome in pairs, but these pairs are sent to different codesat the second level. As a result, the error patterns fromthe first level must come in “matched” pairs that are alsonot detected at the second level. For any particular er-ror configuration going into a block at the second level,there are four preamble configurations and (k− 1) indexconfigurations at the first level that could have caused it.There are (k + 1) undetected error configurations at thesecond level, and the degeneracy factor of four physicalerrors is eight, so the consumed magic states contribute aterm 8(k+1)(k+3)εp

4. Finally, the most likely way thatphysical and encoded errors can occur in conjunction isa logical error on the magic state in question and twophysical errors on the same qubit anywhere, which hasprobability (k + 4)2εlεp

2. Combined, these error terms

reproduce the results in Eqn. (2): E(k+4)×(k+4)2 (εl, εp) =

(k2−1)εl2+8(k2+4k+3)εp

4+(k+4)2εlεp2+(. . .). We drop

terms at higher order because they are found to be negli-gible in optimal protocols. For example, the first optimaltwo-level protocol has parameters k = 8, εl = 3.5×10−5,and εp = 9 × 10−4, where both input types come fromearlier rounds of distillation (Bravyi-Kitaev and Meier-Eastin-Knill, respectively). More details of the numericalsearch are given below.

Continuing this approach, one can show the significant

error terms at level t ≥ 3 are given by

E(k+4)t

t (εl, εp) = (kt − 1)εl2

+2(2t+t−3)(k + 1)(k + 3)t−1εp

(2t)

+(k + 4)t(2(t−2))εlεp

(2(t−1))

+(. . .). (A1)

These terms incorporate degeneracy in error configura-tions and repeated Hadamard measurements. The co-efficients of the second and third terms on the RHSof Eqn. (1) represent physical error configurations andencoded/physical combinations, respectively, and thesegrow rapidly as a function of r. Accordingly, the optimal-protocol search does not advocate the use of three-levelprotocols until the desired output error rate is below10−25, which is beyond the needs of any quantum al-gorithm so far conceived. No four-level protocols werefound to be optimal for output error rates above 10−40,which under practical considerations means they are notlikely to ever be used. The next section considers thesize of multilevel distillation circuits, which can also limittheir usefulness.

Appendix B: Optimal multi-round distillation

The claimed efficiency of multilevel distillation was ex-amined quantitatively with a numerical search for opti-mal multi-round distillation protocols. Each of the pro-tocols is optimal in the sense that, for a given final infi-delity εout, no other sequence requires fewer average inputstates, and, for a given average number of input states,no other protocol achieves lower εout. Note that proba-bility of rejection upon detected error is incorporated byconsidering average cost for distillation when failure-and-repeat steps are included. The protocols plotted in Fig. 3of the main text are just the last round of a distillationsequence. Earlier rounds can be, and usually are, dif-ferent protocols. The search space was constrained suchthat the number of rounds r ≤ 5, number of encoded log-ical qubits k ≤ 20 for all H codes, and multilevel codesare square (k + 4)× (k + 4), etc.

Generally speaking, smaller protocols handle large in-put error rates in early rounds better, while larger mul-tilevel protocols are more inefficient at distillation wheninput error rates are low enough. For example, the proto-cols listed in Table I are the same ones plotted in Fig. 3.Note that when εout is smaller than 10−7, multilevel pro-tocols are most efficient. Should one desire εout < 10−25,level-three protocols have the highest efficiency. Higherlevels of concatenation (up to level five) were part of thesearch, but they were not efficient for εout > 10−40. Theerror-function notation E(·) from the main text is usedto show how the inputs to later rounds of distillation maybe the outputs of an earlier round. This notation neglectsthe total size of the distillers, which must be determinedby using parallel distillation blocks whenever correlatederrors on logical magic states are present.

8

For reference, the best achievable results with priorprotocols are also shown in Table I, in reverse chronolog-ical order of their discovery. The methods are cumulative,so the most recent Bravyi-Haah codes (2012) could alsouse Meier-Eastin-Knill (2012) or Bravyi-Kitaev (2005)distillation, but the oldest Bravyi-Kitaev distillation isalone in its column. The best achievable results witholder protocols are also shown in Fig. 3 for comparison.Accordingly, the multilevel protocols can use any of theabove protocols wherever the numerical search finds do-ing so to be optimally efficient.

The numerical simulation uses error functions E(·) forthe Bravyi-Kitaev (“BK,” [8]) 15-to-1 and Meier-Eastin-Knill (“MEK,” [11]) 10-to-2 protocols. The first threeTaylor-series terms of these functions near ε = 0 are:

EBK(ε) = 35ε3 + 105ε4 + 378ε5 + (. . .); (B1)

EMEK(ε) = 9ε2 − 56ε3 + 160ε4 + (. . .). (B2)

In the numerical search, ε ≤ 0.01, so the first term foreach dominates. The Bravyi-Haah triorthogonal codes(“BH,” [12]) have the same error function as H codes(cf. Eqn. (1) of the main text):

E(k)BH(ε) = E

(k+4)1 (ε, ε) = (3k + 1)ε2 + (. . .). (B3)

This correspondence, combined with the results of Re-ichardt [10], suggests a connection between these codefamilies.

Appendix C: Distilling magic states for T gates withH codes

In addition to distilling Hadamard states|H〉, H codes can also distill the magic state

|A〉 = (1/√

2)(|0〉+ eiπ/4 |1〉), which is used to makethe T = exp(iπ(I − σz)/8) gate. This construction isuseful by itself, but it can also be used to make Toffolimagic states as shown in the next section. State |A〉 is

stabilized by the operator TXT† = (1/√

2)(X + Y), whichis also transversal in H codes. Distillation is performedby encoding |A〉 states as logical qubits, then measuringthe controlled-(TXT†) using |A〉 states at the physicallevel, followed by routine error detection.

Appendix D: Distilling Toffoli magic states

A CSS quantum code [20, 21] necessarily has atransversal CNOT operation. A CSS code with transversalT operation (let us use the shorthand “CSS+T”) will alsohave a transversal controlled-controlled-Z (CCZ) opera-tion, because the latter quantum gate can be decomposedinto CNOT and T (or T†), as shown in Fig. 4 of this supple-mentary information. The CCZ gate is locally equivalentto Toffoli (via Hadamard transforms on the target qubit),

T T

T

S

T+ T+

T+ T+

Z

=

FIG. 4. A controlled-controlled-Z gate, which is locally equiv-alent to a Toffoli, can be decomposed into CNOT and T gates.A CSS+T code has transversal CNOT and T, so it could be usedto distill three-qubit magic states for the Toffoli gate.

so CSS+T codes can distill a magic state for the CCZgate, which is equivalent to distilling Toffoli magic states(see p. 488 of Ref. [4]), assuming Clifford operations arefreely available.

9

− log10(εtarget) − log10(εout) Protocol CML CBH CMEK CBK

4 4.46 EBK(ε0) 17.44 17.44 17.44 17.445 5.14 EMEK(EBK(ε0)) 27.93 27.86 27.86 261.5

6 6.83 E(40)BH (EBK(ε0)) 56.07 56.07 83.99 261.5

7 7.11 E(12×12)2 (EBK(ε0), EMEK(ε0)) 57.38 58.30 83.99 261.5

8 8.06 E(10×10)2 (EMEK(EMEK(ε0)), EMEK(ε0)) 67.52 89.26 139.3 261.5

9 9.08 E(10×10)2 (EMEK(E

(2)BH(ε0)), E

(2)BH(ε0)) 100.3 139.3 139.3 261.5

10 11.1 E(24×24)2 (E

(40)BH (EBK(ε0)), EBK(ε0)) 110.7 179.4 261.7 261.5

11 11.1 —same as above— 110.7 179.4 261.7 261.5

12 12.1 E(24×24)2 (E

(10×10)2 (EBK(ε0), EMEK(ε0)),

EBK(ε0))113.7 187.9 418.0 3923.

13 13.0 E(24×24)2 (E

(10×10)2 (E

(8)BH(EMEK(ε0)))),

EMEK(ε0)), EBK(ε0))120.4 225.6 418.0 3923.

14 14.1 E(24×24)2 (E

(10×10)2 (EMEK(EMEK(ε0)))),

EMEK(ε0)), EBK(ε0))126.9 285.6 419.9 3923.

15 15.0E

(14×14)2 (E

(10×10)2 (EMEK(EMEK(ε0)))),

EMEK(ε0)), E(10)BH (EMEK(ε0)))

158.5 315.5 696.7 3923.

16 16.3 E(24×24)2 (E

(10×10)2 (E

(40)BH (EBK(ε0)),

EMEK(ε0)), EMEK(EMEK(ε0)))187.9 406.2 696.7 3923.

17 17.0 E(22×22)2 (E

(24×24)2 (E

(40)BH (EBK(ε0)),

EBK(ε0)), EMEK(EMEK(ε0)))195.5 529.5 696.7 3923.

18 18.0E

(20×20)2 (E

(24×24)2 (E

(38)BH (EBK(ε0)),

EBK(ε0)), EMEK(E(40)BH (ε0)))

239.8 574.1 1260. 3923.

19 19.5E

(24×24)2 (E

(24×24)2 (E

(40)BH (EBK(ε0)),

EBK(ε0)), E(40)BH (EBK(ε0)))

272.1 574.1 1260. 3923.

20 20.0E

(24×24)2 (E

(24×24)2 (E

(30)BH (EBK(ε0)),

EBK(ε0)), E(40)BH (EBK(ε0)))

273.3 574.1 1260. 3923.

21 21.6E

(24×24)2 (E

(24×24)2 (E

(10×10)2 (EBK(ε0),

EMEK(ε0)), EBK(ε0)), E(40)BH (EBK(ε0)))

275.1 575.9 1260. 3923.

22 22.0E

(24×24)2 (E

(20×20)2 (E

(10×10)2 (EBK(ε0),

EMEK(ε0)), EBK(ε0)), E(40)BH (EBK(ε0)))

278.0 604.3 1308. 3923.

23 23.3E

(24×24)2 (E

(24×24)2 (E

(10×10)2 (E

(8)BH(EMEK(ε0)),

EMEK(ε0)), EBK(ε0)), E(40)BH (EBK(ε0)))

281.9 652.3 2090. 3923.

24 24.2E

(24×24)2 (E

(24×24)2 (E

(10×10)2 (E

(6)BH(EMEK(ε0)),

EMEK(ε0)), EBK(ε0)), E(12×12)2 (EBK(ε0), EMEK(ε0)))

287.9 731.5 2090. 3923.

25 25.1 E(16×16×16)3 (E

(22×22)2 (E

(10×10)2 (EMEK(EMEK(ε0)),

EMEK(ε0)), EBK(ε0)), EMEK(EMEK(ε0)))295.7 853.1 2090. 3923.

26 26.1 E(16×16×16)3 (E

(14×14)2 (E

(10×10)2 (EMEK(EMEK(ε0)),

EMEK(ε0)), EBK(ε0)), EMEK(EMEK(ε0)))311.5 914.0 2090. 3923.

27 27.1E

(16×16×16)3 (E

(24×24)2 (E

(10×10)2 (EMEK(E

(2)BH(ε0)),

EMEK(ε0)), E(6)BH(EMEK(ε0))), EMEK(EMEK(ε0)))

333.3 947.5 2100. 3923.

28 28.1 E(16×16×16)3 (E

(18×18)2 (E

(10×10)2 (EMEK(E

(2)BH(ε0)),

EMEK(ε0)), EMEK(EMEK(ε0))), EMEK(EMEK(ε0)))355.6 1015. 2181. 3923.

29 29.3 E(16×16×16)3 (E

(18×18)2 (E

(10×10)2 (E

(40)BH (EBK(ε0)),

EMEK(ε0)), EMEK(EMEK(ε0))), EMEK(EMEK(ε0)))363.7 1125. 3483. 3923.

30 30.7 E(16×16×16)3 (E

(24×24)2 (E

(24×24)2 (E

(40)BH (EBK(ε0)),

EBK(ε0)), EMEK(EMEK(ε0))), EMEK(EMEK(ε0)))369.3 1301. 3483. 3923.

31 31.0 E(16×16×16)3 (E

(20×20)2 (E

(24×24)2 (E

(40)BH (EBK(ε0)),

EBK(ε0)), EMEK(EMEK(ε0))), EMEK(EMEK(ε0)))376.5 3923.

32 32.4E

(16×16×16)3 (E

(24×24)2 (E

(24×24)2 (E

(40)BH (EBK(ε0)),

EBK(ε0)), EMEK(E(2)BH(ε0))), EMEK(EMEK(ε0)))

411.5 3923.

33 33.0E

(14×14×14)3 (E

(20×20)2 (E

(24×24)2 (E

(38)BH (EBK(ε0)),

EBK(ε0)), EMEK(E(2)BH(ε0))), EMEK(EMEK(ε0)))

427.3 3923.

34 35.2E

(16×16×16)3 (E

(24×24)2 (E

(24×24)2 (E

(40)BH (EBK(ε0)),

EBK(ε0)), E(40)BH (EBK(ε0))), EMEK(EMEK(ε0)))

459.4 58838.

35 35.2 —same as above— 459.4 58838.

10

36 36.1E

(24×24×24)3 (E

(24×24)2 (E

(24×24)2 (E

(30)BH (EBK(ε0)),

EBK(ε0)), E(40)BH (EBK(ε0))), E

(40)BH (EBK(ε0)))

470.8 58838.

37 37.3E

(24×24×24)3 (E

(24×24)2 (E

(24×24)2 (E

(12×12)2 (EBK(ε0),

EMEK(ε0)), EBK(ε0)), E(40)BH (EBK(ε0))), E

(40)BH (EBK(ε0)))

471.0 58838.

38 39.4E

(24×24×24)3 (E

(24×24)2 (E

(24×24)2 (E

(10×10)2 (EBK(ε0),

EMEK(ε0)), EBK(ε0)), E(40)BH (EBK(ε0))), E

(40)BH (EBK(ε0)))

472.6 58838.

39 39.4 —same as above— 472.6 58838.TABLE I: Optimal distillation protocols identified by numerical search.Protocols are specified using the error functions E(·) to indicate wheninputs to one round are the outputs of another distillation circuit. Thedata for CBH and CMEK are obtained from Ref. [12].