The Advantages of Qudit Fault-Tolerance

The advantages of qudit fault-tolerance

Earl T. Campbell (Sheffield) Quantum Error Correction 2014 Zurich

Background image: Computation Cloud installation piece Libby Heany http://earltcampbell.com/research/ Earl WHY QUDITS? Campbell

 Monz et al. Phys. Rev. Lett. 106 130506 (2011)

 Smith et al. Phys. Rev. Lett. 111, 170502 (2013) Anderson et al. arXiv:1410.3891 (2014) Earl Campbell

Setun Earl Campbell

Setun

“Perhaps the prettiest number system of all is the balanced ternary notation” Donald Knuth Earl OVERVIEW Campbell

A contemporary approach to fault-tolerant quantum computing

STORAGE MAGIC STATES COMPILING ALGORITHMS PROFIT

Use QEC code Prepare magic Compose Implement to reduce states and finite gate set ! " # quantum noise. inject extra to produce any computation, gates. gate.  Earl ROUTES TO UNIVERSALITY Campbell

fault-tolerant ~ transversal M M M M... L ⇠ ⌦ ⌦

Storage Magic States

CODE 1 CODE 2 Efficient, high-threshold, Allows fault-tolerant allows fault-tolerant implementation of a implementation of non-Clifford gate, (most) Clifford group gates. e.g. pi/8 gate e.g. Toric code e.g. Reed-Muller codes Earl ROUTES TO UNIVERSALITY Campbell

SUBSYSTEM CODES + GAUGE FIXING Only 1 code. Potentially fewer resources needed. But must also allow fault-tolerant non-Clifford gate.

Storage-Magic States e.g. gauge colour codes, or gauge variants of Reed-Muller codes

Many other alternative, but all rely on these exotic codes A. Paetznick and B. W. Reichardt, Phys. Rev. Lett. 111, 090505 (2013). H. Bombin et. al., New J. Phys. 15, 055023 (2013) J. T. Anderson et. al., Phys. Rev. Lett. 113, 080501 (2014). T. Jochym-O’Connor and R. Laflamme, Phys. Rev. Lett. 112, 010505 (2014). Earl OVERVIEW Campbell

Qudit Toric Code Qudit Magic New J. Phys. Phys. Rev. X Phys. Rev. Lett 16 063038 (2014) 2 041021 (2012) 113 230501 (2014)

PHYSICAL REVIEW X 2, 041021 (2012) Enhanced fault-tolerant quantum computing in d-level systems Enhanced fault-tolerant quantum computing in d-level systems Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes Earl T. Campbell1, 2 1, 2 1 Earl T. Campbell 1, 2 2 Department of Physics & Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom. Earl T. Campbell, * Hussain Anwar, and Dan E. Browne 1 2DahlemDepartment Center of Physics for Complex & Astronomy, Quantum University Systems, Freie of Sheffield, Universit Sheffield,ät Berlin, S3 14195 7RH, Berlin, United Germany.Kingdom. 1Dahlem Center for Complex Quantum Systems, Freie Universita¨t Berlin, 14195 Berlin, Germany 2Dahlem Center for Complex Quantum Systems, Freie Universitat¨ Berlin, 14195 Berlin, Germany. 2Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom Error correcting codes protect quantum information and formthebasisoffault-tolerantquantumcomputing. (Received 29 May 2012; revised manuscript received 19 July 2012; published 27 December 2012) LeadingError correctingproposals forcodes fault-tolerant protect quantum quantum information computation and formrequire the codes basis withof fault-tolerant an exceedingly quantum rare computing.property, a transversalLeading proposals non-Clifford for fault-tolerant gate. Codes quantum with the computation desired property require are codes presented with for and exceedingly-level, qudit, rare systems property, with a We propose families of protocols for magic-state distillation—important components of fault-tolerance primetransversald.Thecodesuse non-Cliffordn gate.= d Codes1 qudits with and the desiredcan detect property up to ared/ presented3 errors. We for quantifyd-level, qudit, the performance systems with of schemes—for systems of odd prime dimension. Our protocols utilize quantum Reed-Muller codes with prime d. The codes use n = d 1 qudits and can detect up to ⇠ d/3 errors. We quantify the performance of these codes for one approach to quantum computation known as magic-state⇠ distillation. Unlike prior work, we transversal non-Clifford gates. We find that, in higher dimensions, small and effective codes can be used findthese performance codes for one is alwaysapproach enhanced to quantum by increasing computationd. known as magic-state distillation. Unlike prior work, we Fast decoders for qudit topological codes that have no direct analogue in qubit (two-dimensional) systems. We present several concrete protocols, find performance is always enhanced by increasing d. including schemes for three-dimensional (qutrit) and five-dimensional (ququint) systems. The five- PACS numbers: 03.67.Ac,03.67.Lx,03.67.Pp PACS numbers: 03.67.Ac,03.67.Lx,03.67.Pp dimensional protocol is, by many measures, the best magic-state-distillation scheme yet discovered. It 1,2,5 3,5 4 Hussain Anwar , Benjamin J Brown , Earl T Campbell and excels both in terms of error threshold with respect to depolarizing noise (36:3%) and the efficiency 2 Quantum error-correction stores information in a subspace eral quantifiers of performance for magic state distillationus- Dan E Browne measure known as yield, where, for a large region of parameters, it outperforms its qubit counterpart by Quantum error-correction stores information in a subspace eral quantifiers of performance for magic state distillation us- 1 of a larger physical Hilbert space and is an efficient method of ing these codes [31, 32]. CAB found marked improvement Department of Mathematical Sciences, Brunel University, Uxbridge, Middlesex UB8 3PH, UK many orders of magnitude. of a larger physical Hilbert space and is an efficient method of ing these codes [31, 32]. CAB found marked improvement 2 protecting quantum information from noise. Repeated mea- over comparable qubit codes for modest sizes d =3, 5.How- Department of Physics and Astronomy, University College London, Gower Street, London protecting quantum information from noise. Repeated mea- over comparable qubit codes for modest sizes d =3, 5. How- WC1E 6BT, UK DOI: 10.1103/PhysRevX.2.041021 Subject Areas: Quantum Information surements and error-corrections keep the information from ever, in even larger dimensions (d>5)performanceagain 3 d>5 Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London, London driftingsurements too and far error-corrections out of the error-correction keep the information subspace, from also declined.ever, in even This larger fall in dimensions performance ( is peculiar,) performance especially againin SW7 2AZ, UK Despite these obstacles, fault-tolerant universal quan- calleddrifting the too codespace. far out of For the robust error-correction quantum computation, subspace, also we lightdeclined. of other This results fall showingin performance qudit toric is peculiar, codes have especially thresholds in 4 I. INTRODUCTION Dahlem Center for Complex Quantum Systems, Freie Universitát Berlin, D-14195 Berlin, tum computing is possible [1]. One particularly success- mustcalled be the able codespace. to perform For gates robust without quantum leaving computation, a protected we increasinglight of other with results system showing dimension qudit toric[33–35]. codes Here have we thresholds present Germany The central challenge of implementing scalable quan- ful approach, known as state injection, is to achieve codespacemust be able orto amplifying perform gates existing without errors. leaving Fault-tolerance a protected MSDincreasing protocols withwith system commensurate dimension [33–35]. improvements Here we to presentthresh- E-mail: [email protected] tum computing is to protect quantum systems against noise universality by augmenting the fault-tolerant operations iscodespace straightforward or amplifying for a limited existing set errors. of gates, Fault-tolerance the so-called oldsMSD and, protocols more importantly,with commensurate an unexpected improvements improvement to thresh- to and decoherence while retaining the capacity to perform is straightforward for a limited set of gates, the so-called olds and, more importantly, an unexpected improvement to Received 21 February 2014 with a supply of many copies of a suitable ancillary transversal gates of the code. Unfortunately, severe con- efficiency that is not present in toric codes. Fault-tolerantClif- Accepted for publication 15 April 2014 computation. Quantum error correction and fault-tolerant resource state. While methods for the direct preparation straintstransversal existgates [1–4] of that the mean code. such Unfortunately, direct approaches severe canno con-t fordefficiency gates [36]that is can not be present provided in toric by toric codes. codes, Fault-tolerant and our proto- Clif- Published 17 June 2014 techniques provide a solution to this problem, and a variety of sufficiently noise-free protected resource states have providestraints existgates [1–4] sufficient that meanfor universal such direct quantum approaches computation cannot. colsford extend gates [36] this can to a be universal provided set by of toric fault-tolerant codes, and gates; our proto-with of constructions for fault-tolerant quantum computation New Journal of Physics 16 (2014) 063038 been proposed [1], a particularly elegant solution can be Rather,provide we gates must sufficient rely on for additional universal techniques quantum to computation. implement bothcols extend components this to becoming a universal more set of effective fault-tolerant in higher gates; dimen- with have been proposed [1–4]. In all these schemes, a delicate doi:10.1088/1367-2630/16/6/063038 provided by distillation techniques, outlined in Fig. 1, furtherRather, fault-tolerant we must rely gates. on additional techniques to implement sions,both components they pose a powerful becoming combination. more effective Such in gains higher will dimen- also balance must be maintained between coherently manipu- where many noisy copies of a resource state can be furtherOne route fault-tolerant to universality gates. is to prepare high-fidelity resource applysions, to they qudit pose analogs a powerful of the combination. “magic-state free” Such fault-tole gains willrance also lating the encoded system while preserving the protected Abstract distilled to arbitrary fidelity by using only error-protected statesOne and route then to use universality state-injection is to prepare to convert high-fidelity the resource resourcestate schemesapply to qudit [18–23]. analogs of the “magic-state free” fault-tolerance subspace and prohibiting the proliferation of errors. For Qudit toric codes are a natural higher-dimensional generalization of the well- operations while preserving the error threshold of the intostates a andfault-tolerant then use state-injection gate [5–11]. Reduction to convert the of noise resource in the statese schemes [18–23]. example, for schemes built on stabilizer codes [5], trans- We consider an extended class of quantum Reed-Muller model. resourceinto a fault-tolerant states, sometimes gate [5–11]. known Reduction as magic of states noise [8],in these re- We consider an extended class of quantum Reed-Muller studied qubit toric code. However, standard methods for error correction of the versal gates have the desired properties, while, in topologi- codes that are constructed in terms of polynomial functions, quiresresource extensive states, distillation sometimes methods known as demanding magic states that the [8], m re-a- codes that are constructed in terms of polynomial functions, qubit toric code are not applicable to them. Novel decoders are needed. In this cal systems, topologically protected braiding operations whereas the work of CAB only considered linear functions. jorityquires of extensive a quantum distillation computer methods is a dedicated demanding magic that state the fac- ma- whereas the work of CAB only considered linear functions. paper we introduce two renormalization group decoders for qudit codes and [2] provide the logical gates. While much work in quantum Here we see that higher degree polynomials can be used to toryjority [12, of a13]. quantum Due to computer the significant is a dedicated resource magic overhead, state max- fac- Here we see that higher degree polynomials can be used to analyse their error correction thresholds and efficiency. The first decoder is a computation has focused on qubits (two-level systems), it construct codes with the desired transversality properties. Fur- More copies imizingtory [12, efficiency 13]. Due of to these the significant protocols resource is of paramount overhead, impor- max- construct codes with the desired transversality properties. Fur- generalization of a ‘hard-decisions’ decoder due to Bravyi and Haah is known that, for any prime d, effective codes exist for thermore, the polynomial degree can grow with the qudit di- lower fidelity tance,imizing and efficiency recently ofmany these improvements protocols is have of paramount been made impor- [14– storing d-level quantum systems [5–7]. Thus, qudit sys- (arXiv:1112.3252). We modify this decoder to overcome a percolation effect 16].tance, One and couldrecently try many to circumvent improvements this haveoverhead been by made explor- [14– which limits its threshold performance for many-level quantum systems. The tems are also candidates for scalable fault-tolerant quan- 16]. One could try to circumvent this overhead by explor- 3.0 1.0 tum computation. ing one of many other ways to achieve universality [17–23]. 3 fig (a) fig (c) ‘ ’ ing one of many other ways to achieve universality [17–23]. 7 0.8 second decoder is a generalization of a soft-decisions decoder due to Poulin However, all these proposals have to sacrifice some error cor- 2.5 2 In many approaches, the protected unitary gates are a However, all these proposals have to sacrifice some error cor- 17 and Duclos-Cianci (2010 Phys. Rev. Lett. 104 050504), with a small cell size to recting capabilities, and so are only viable when physical op- 5 0.6 11 13 subset of the so-called Clifford group. It is known that the 2.0 13 5 optimize the efficiency of implementation in the high dimensional case. In each erationsrecting capabilities, are much less and noisy so are (such only as viable was explicitly when physical shown op- in 11 19 0.4 stabilizer operations (comprising Clifford unitaries as well 17 23 7 erations are much less noisy (such as was explicitly shown in 1.5 3 case, we estimate thresholds for the uncorrelated bit-flip error model and provide Ref. [19]). Except for Shor’s method [17], these alternative 0.2 2 as preparation and measurements in the computational Fewer copies Ref. [19]). Except for Shor’s method [17], these alternative a comparative analysis of the performance of both these approaches to error routes require codes with a rare property, and such codes also 1.0 d 0.0 basis) can be classically simulated efficiently [5,6,8] and 2 higher fidelity playroutes a fundamental require codes role with in amost rare magic property, state and distillation such codes (herein also fig (b) d correction of qudit toric codes. that, on their own, they are not universal for quantum Bravyi-Haah limit MSD)play a fundamentalprotocols. Specifically, role in most these magic codes state have distillation as a transver (herein- 1.5 computation. Furthermore, several theorems have shown FIG. 1. An outline of a single round of magic-state-distillation MSD) protocols. Specifically, these codes have as a transversal gate the U⇡/8 phase gate, which is special as it is outside 1 Keywords: topological error correcting codes, toric code, qudits, thresholds, [9–12] that, in general, there is a tension between providing 10 10 3 10 5 10 7 10 9 d,logscale protocol. Within many architectures of fault-tolerant quantum thesal Cliffordgate the groupU⇡/8 phase yet still gate, closely which related is special to it. as it is outside renormalization group decoders protection against generic noise and achieving universal computing, a large proportion of the device is committed to these theIn Clifford almost groupevery route yet still to closely fault-tolerance, related to these it. exotic codes quantum computing. magic-state factories. Each attempt uses n copies of a state and In almost every route to fault-tolerance, these exotic codes FIG. 1. The performance of our codes at MSD. (a) shows the when successful outputs a state E n . For n given suc- emerge as pivotal components. Here, we tackle the problem FIG. 1. The performance of ourQRM codes at MSD. (a) shows the 5 These authors contributed equally to this paper. 0 emerge as pivotal components. Here, we tackle the problem efficiency metric (the smallerQRM the better) for various prime d. (b) cessful attempts, the output states are/ usedð asÞ inputs into the next of designing and improving analogous codes in the qudit set- efficiency metric (the smaller the better) for various prime d. (b) *[email protected] of designing and improving analogous codes in the qudit set- Two curves which fluctuates between, shown for much larger d (on iteration. Within the magic-state model, the completely positive ting of using d-level elementary systems. In this setting, qu- alog-scale).Two curves which(c) The fluctuates depolarizing between, noise shown threshold for much below larger whichd the(on Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. ting of using d-level elementary systems. In this setting, qu- a log-scale). (c) The depolarizing noise threshold below which the Published by the American Physical Society under the terms of map, E, is composed of a sequence of Clifford unitaries and Pauli dit error-correction [27–29] has been long known, but only states are distillable. Raised bars show the theoretical maxima where Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal dit error-correction [27–29] has been long known, but only states are distillable. Raised bars show the theoretical maxima where the Creative Commons Attribution 3.0 License. Further distri- measurements. This figure illustrates a protocol where n 4, for much later were analogs of U⇡/8 phase gate characterized [30] distillation becomes impossible [24, 25]. Where multiple maxima citation and DOI. ¼ distillation becomes impossible [24, 25]. Where multiple maxima bution of this work must maintain attribution to the author(s) and example, the ququint, d 5, protocol that we discuss through- andmuch codes laterwere discovered analogs with of U these⇡/8 phase as transversal gate characterized gates [31]. [30] appear (see d =7), not all magic states Mµ are equivalent (see ¼ appear (see d =7), not all magic states |Mµi are equivalent (see New Journal of Physics 16 (2014) 063038 the published article’s title, journal citation, and DOI. out the article. and codes discovered with these as transversal gates [31]. Ref. [26] and the Supplemental Material). Campbell, Anwar, and Browne (herein CAB) analyzed sev- Ref. [26] and the Supplemental Material). | i 1367-2630/14/063038+37$33.00 © 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft Campbell, Anwar, and Browne (herein CAB) analyzed sev-

2160-3308=12=2(4)=041021(20) 041021-1 Published by the American Physical Society

Benjamin Dan Hussain Brown Browne Anwar Earl INTRODUCING QUDITS Campbell

Define a basis 0 , 1 ,...,p 1 with all arithmetic modulo p. { } Imagine states as notches on a clock face.

0 | i Pauli-group : P 4 1 generators X n = n +1 | i | i Z|ni = |!n n i X | i | i  i2⇡/p where ! = e

3 2 | i | i

Clifford group is normaliser of so C C† = C P P P Overcomplete set of generators n = ↵n + 1 nm ↵, H n = pp m ! m X | i | ↵n+n2i | i | i n = ! n CX n m = n m + n Z↵,| i | i | i| i | i|P i Assuming p is an odd prime! Earl QUDIT TORIC CODES Campbell

The toric code straightforwardly generalizes though we have to adjust the code slightly

Stabilizers Z are still 4-body terms but “daggerized” Z Z† Z† plaquettes

X† X X† X stars Earl QUDIT TORIC CODES Campbell

The toric code straightforwardly generalizes though we have to adjust the code slightly

Logical operators are still closed loops (in the homological sense)

Z Z Z Z Z Z ZL Earl QUDIT TORIC CODES Campbell

The toric code straightforwardly generalizes though we have to adjust the code slightly X Logical operators X are still closed loops (in the homological sense) X the shortest loop is length X in a code of 2 L 2 qudits.

X They are [[2 L 2 , 2 ,L ]] codes.

XL Earl DECODERS Campbell

MINIMUM WEIGHT PM BROOM RENORMALISATION

S.Bravyi, J, Haah G. Duclos-Cianci, D. Poulin., Phys. Rev. Lett. 111, 200501 Phys. Rev. Lett. 104, 050504 Earl RESULTS Campbell

Decoding and Thresholds Thresholds improve with p; Thresholds follow ~69% of the hashing bound threshold; We need to tweak broom to get this.

Similar results in: threshold (%) threshold  G. Duclos-Cianci, D. Poulin., Phys. Rev. A 87 062338  R. Andrist, J. Wootton, H. Katzgraber., arXiv:1406.5974 dimension p broom Advantage persists in when accessing for noisy stabiliser enhanced-broom measurement. Thresholds up to 8% renormalisation  F. Watson, H. Anwar, D. Browne arXiv:1411.3028 69% hashing bound threshold  H. Anwar, et al New J. Phys. 16 063038 (2014)  Magic States Pt. 1 Implementing non-Clifford gates Earl NON-CLIFFORD GATES Campbell

Good examples of non-Clifford gates

Qubit Qutrit (p=3) Qudits (p>3) n n3 U n = ⌫ n V n = ⌧ n n M n = ! n | i | i | i | i 2⇡ | i | i 2⇡ 2⇡ with ⌫ =exp i with ⌧ =exp i with ! =exp i 23 32 p   

 M. Howard, J. Vala., Phys. Rev. A 86, 022316 (2012) E. Campbell, H. Anwar, D. Browne., Phys. Rev. X 2 041021 (2012) Earl NON-CLIFFORD GATES Campbell

Good examples of non-Clifford gates

Qubit Qutrit (p=3) Qudits (p>3) are CUBIC gates n n3 U n = ⌫ n V n = ⌧ n n M n = ! n | i | i | i | i 2⇡ | i | i 2⇡ 2⇡ with ⌫ =exp i with ⌧ =exp i with ! =exp i 23 32 p   

Clifford hierarchy 3rd level Conjugates Pauli group to Clifford group

M M † = P C

enables gate-injection using resource of magic-states CM | i | i where 1 n3 m M = M + = ! n X Z m C = MX M † | i | i p | i M † | i p n X | i Earl MAGIC STATE DISTILLATION Campbell

i projects onto Given an [[n,k,d]] code with transversal M M | CODE 2 for noise rate ✏ =1 M ⇢ M we find subspace h | | i noisy A threshold: if ✏ < ✏ ⇤ then ✏out < ✏ 1 d Exponential: ✏out

 S. Brayvi and A. Kitaev., Phys. Rev. A 71, 022316 (2005) E. Campbell, H. Anwar, D. Brown., Phys. Rev. X 2 041021 (2012) Earl TRIORTHGONALITY Campbell

G0 r ~0 = 1 u Define a matrix G = Logical states L ppr | i u span(G ) | i G1 k 2 0 ✓ ◆ 1 P n m~ L = ppr u + G1m~ | i u span(G0) | i 2 P Define stabilizer generators X [ ~v ] for all ~v G X[~v]=Xv1 Xv2 ...Xvn 0 logical operators X [ ~v ] for all ~v 2 G ⌦ ⌦ 2 1 Definition

A matrix/code is said to be triorthgonal if for all v, v0,v00 G 2 0 (mod p) ,v G whenever v3 = 2 0 j otherwise, vjvj0 vj00 =0 (modp) v = v0 = v00 j (a =0 (modp) ,v G1 j X 6 2 Theorem P A triorthgonal code has a transversal non-Clifford U, V or M (upto a Clifford)

qubits  S. Brayvi and J. Haah., Phys. Rev. A 86, 052329 (2012) qudits  unpublished / in preparation Earl TRIORTHGONALITY Campbell

Triorthogonal codes Finding the exotic codes

Reed-Muller codes are [[ p m 1 , 1 , ?]] codes where m is the order. Simplest qubit triorthgonal code in the “Reed-Muller” family is order 4.

v 111111110000000 n v0 111100001111000 U n = ⌫ n 0 1 | i | i v00 110011001100110 2⇡ B 101010101010101C ⌫ =exp i B C 23 B 111111111111111C B C  @ A v v0 v00 =( 110000000000000) · · v v0 v00 = 2 = 0 (mod 2) j · j · j j X Definition A matrix/code is said to be triorthgonal if for all v, v0,v00 G 2 0 (mod p) ,v G whenever v3 = 2 0 j otherwise, vjvj0 vj00 =0 (modp) v = v0 = v00 j (a =0 (modp) ,v G1 j X 6 2 P Earl QUTRITS Campbell

Triorthogonal codes Finding the exotic codes

Simplest qudit “Reed-Muller” code for dimension 3

00111222 V n = ⌧ n n 12012012 | i | i 2⇡ 0 1 with ⌧ =exp i 11111111 32 @ A  an [[8 , 1 , 2]] code so order 2 Compared to 15 qubit: 1. smaller code; 2. worse distance; 3. worse overhead (gamma); 4. threshold? Earl QUTRITS: THRESHOLDS Campbell

In higher dimensions, Let us look a slice of state space ✏ =1 M ⇢ M (which we can always project onto) h | | i doesn’t uniquely identify the state. We parameterize by z =tr[M⇢]

Distillable 0.6 0.5 ✏dep⇤ =0.363Physical 0.4 ✏dep⇤ =0.211 0.5 ✏⇤ =0.311 Bound 0.4 0.3 ✏⇤ =0.200 0.3 Stabilizer 0.2 0.2 Re(z) 0.1 0.1 Output error rate Output error 0 0 rate Output error 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 Input error rate Input error rate Im(z)

 Veitch et. al. New Journal of Physics 14 113011 (2012) Earl QUDITS IN BIG DIMENSIONS Campbell

Triorthogonal codes Finding the exotic codes

Simplest qudit “Reed-Muller” code

for dimension 5 3 M n = !n n 1234 | i | i 2⇡ with ! =exp i 1111 p ✓ ◆  for dimension 7 123456 111111 ✓ ◆ for dimension 11 1 2 3 4 5 6 7 8 9 10 111111111 1 ✓ ◆ Gives a triorthonormal family of order 1 Reed-Muller codes. [[p 1, 1, 2]] Earl PERFORMANCE Campbell

Threshold Overhead 1 5 ⇤ dep ✏ 0.75 4

0.5 3 Gamma Threshold Threshold 0.25 2

0 1 2 3 5 7 11 13 17 2 3 5 7 11 13 17 19 23 Qudit dimension p Qudit dimension p

remember N A log (✏ )  ✏ target  Magic States pt 2 Beyond linear functions Earl HIGHER DEGREE Campbell

Reed-Muller codes in more detail

stick to first-order codes [[(pm 1), 1, ?]] define generating matrix in terms of functions

f (1) f (2) ... f (p 1) 1 1 1 f2(1) f2(2) ... f2(p 1) G0 0 . 1 G = = . G1 B C ✓ ◆ B f (1) f (2) ... f (p 1) C B r r r C B 11... 1 C B C @ A

earlier we had only 1 function f1(x)=x

a degree-r Reed-Muller code to has generating functions 2 r f1(x)=x, f2(x)=x ,...fr(x)=x Earl KEY RESULT! Campbell

Reed-Muller codes in more detail

a degree-r Reed-Muller code to has generating functions 2 r f1(x)=x, f2(x)=x , fr(x)=x Result

A degree-r Reed-Muller code is triorthgonal if r<(p 1)/3 Furthermore, it can detect up-to r errors.

1. consider a triple-product from G0 2. for prime numbers we know q g(x)=fa(x)fb(x)fc(x)=x 0 q =0 (modp 1) xq (mod p)= 6 q = a + b + c x (p 1 q =0 (modp 1) X

3. triorthgonality ensured if a, b, c r we have a + b + c

Overhead Threshold

5 1

⇤ dep 0.75 4 ✏ no cloning limit for QEC 0.5 3  Gamma Threshold

0.25 2

0 1 2 3 5 7 11 13 17 2 3 5 7 11 13 17 19 23 dimension p dimension p Earl PERFORMANCE Campbell

Overhead Threshold

5 1

⇤ dep 0.75 4 ✏

0.5 3 Gamma Threshold

0.25 2

0 1 2 3 5 7 11 13 17 2 3 5 7 11 13 17 19 23 dimension p dimension p

 I. Bengtsson, K. Blanchfield, E. Campbell, M. Howard J. Phys. A: Math. Theor. 47 455302 (2014) appendix of E. Campbell. Phys. Rev. Lett 113 230501 (2014) Earl GRAND COMPARISON Campbell

Qudit QRM Qubit QRM Bravyi-Haah Meier et. al.

1 Better

0.75 17 11 13 0.5 5 7 Noise threshold Noise 3 0.25 Worse

0 1.5 1.875 2.25 2.625 3

Bravyi-Haah limit Overhead measured by gamma  S. Brayvi and J. Haah., Phys. Rev. A 86, 052329 (2012) A. Meier, B. Eastin, and E. Knill., QIC 13, 195 (2013) Earl BALANCED INVESTMENT Campbell Earl MULTI-LEVEL DISTILLATION Campbell

raw

 C. Jones., Phys. Rev. A 87, 042305 (2013) Earl CIRCUIT COMPLEXITY Campbell

Value? Thanks to D. Browne, H. Anwar, M. Howard, F. Watson, I. Bengtsson, K. Blanchfield, and THANK YOU! Stooge slide 1. Hey Earl, can you show that M is in the 3 level of Clifford hierarchy? 2. Can you say more about how triorthgonality entails transversality of a non-Clifford? 3. Why does the distance increase for higher degree Reed-Muller code? 4. Hang on, what is a Reed-Muller codes? 5. Can I see more plots please? 6. OK, Gamma is high and thresholds are high, but whats about yields? Background image: 7. Aren’t colour codes fun, what about gauge colour Computation Cloud installation piece codes? Libby Heany Earl NON-CLIFFORD PROOF Campbell

1. You want me to show that M is in the 3 level of Clifford hierarchy?

M M † = P C

2. You want me to show that triorthgonality entails M transversality?

1 Qudits (p>3) 0 = u | Li p | i n3 p M n = ! n u span[G0] | i | i 2 X 2⇡ so clearly ML u = u , u span[G0]= u = ML 0L = 0L with ! =exp i | i | i 8 2 | i ) | i | i p  µ µ u3 M u = M u = ! j j L| i | i so for a transversal gate n P O µ µ u3 3 µ µ require M u = M u = ! j j u entails µ u =0 (modp) ML = M M ...Mµ L| i | i | i j ⌦ ⌦ n P j we demand that O X 3 we write u = v where is some subset of rows from G0 M m = !m m U L L L v | i | i X2U for a flavour of the proof let 3 3 us look at just j uj = j( v vj) 2U ML 0L = 0L | i | i = v v v j vjvj0 vj00 P P 2PU 02U 002U P P ⇣zeroP for triorthgonalP matrix⌘ Earl NON-CLIFFORD PROOF Campbell

3. You want me to show that code distance scales with r?

Want to find minimum weight Z[u] such that [ Z [ u ] ,X [ v ]] = 0 for all v G0 2 u v [Z[u],X[v]] = ! j j j we write v = f(1),f(2),...f(p 1) , u = h(1),h(2),...h(p 1) P { } { } [Z[u],X[v]] = 0 f(x)h(x)=0 (modp) () x let us assume for brevity h(x)=xm X m+t therefore require x =0 (modp) for all t r x  Recall: for prime numbersX we know

0 q =0 (modp 1) xq (mod p)= 6 SO,… p 1 q =0 (modp 1) m p 1 r x ( X but low degree functions can only have a small number of zeros, which tells us wt[Z(u)] p 1 m Therefore wt[Z(u)] r Earl MORE ON REED-MULLER CODES Campbell

4. Hang on, what is a quantum Reed-Muller code again?

for order m we have a [[ p m 1 , 1 ,d ]] code m consider functions f : F / 0 Fp p { } !

then G 0 has rows corresponding to different functions and columns evaluate the function at different values

so example if p=2 and m=3, with 3 different function f1, f2, f3

f1(0, 0, 1) f1(0, 1, 0) f1(1, 0, 0) f1(1, 1, 0) f1(1, 0, 1) f1(0, 1, 1) f1(1, 1, 1) G = f (0, 0, 1) f (0, 1, 0) f (1, 0, 0) f (1, 1, 0) f (1, 0, 1) f (0, 1, 1) f (1, 1, 1) 0 0 2 2 2 2 2 2 2 1 f3(0, 0, 1) f3(0, 1, 0) f3(1, 0, 0) f3(1, 1, 0) f3(1, 0, 1) f3(0, 1, 1) f3(1, 1, 1) @ A these functions can again be polynomials of x1, x2, x3,

f1(x1,x2,x3)=x1 0011101 For example f2(x1,x2,x3)=x2 gives G = 0101011 0 0 1 f3(x1,x2,x3)=x3 1101011 @ A Earl PERFORMANCE OF DIMENSION 5 CASE Campbell

5/6. More plots please, yes of course….

dimension 3 dimension 5

-5 target -5 (1a) 3(2) 10 (2a) 5(1) 10 target QRM -10 QRM -10 10 10 10-15 10-15 Y

1 1 Y ield, Y ield, Y 10-5 10-5

-10 10-10 10 0 0.1 0.1 0.2 in in 0.2 0.3 1 (1b) cross-section 1 (2b) cross-section -2 -2 10 3(2) 10 5(1) QRM -8 QRM -8 -4 -4 target =10 10 target =10 10 -6

ield, Y -6 Yield, Y Yield,

10 Y 10 10-8 10-8

0.05 0.1 0.15 0.2 0 0.1 0.2 0.3 in in