Software Pauli Tracking for Quantum Computation

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by  Alexandru Paler

Software Pauli Tracking for Quantum Computation
1 2 2 1
Alexandru Paler Simon Devitt Kae Nemoto Ilia Polian
1Faculty of Informatics and Mathematics 2National Institute of Informatics
University of Passau 2-1-2 Hitotsubashi, Chiyoda-ku
Innstr. 43, D-94032 Passau, Germany Tokyo, Japan
alexandru.palerilia.polian @uni-passau.de devittnemoto @nii.ac.jp
{ | } { | }
Abstract—The realisation of large-scale quantum computing QECC schemes, allowingfor scalable,large-scaleinformation
4 isnolongersimplyahardwarequestion.Therapiddevelopment processing [12], [13]. However, as quantum teleportation is
1 of quantum technology has resulted in dozens of control and inherently probabilistic the direction of qubit rotations is ran-
0 programming problems that should be directed towards the dom.Thisrandomnesscanbecorrectedviaatechniqueknown
2 classicalcomputerscienceandengineeringcommunity.Onesuch
as Pauli tracking. Pauli tracking operates by constructing a
n problem is known as Pauli tracking. Methods for implementing classical record of each teleportation result and reinterprets
a quantum algorithms that are compatible with crucial error later results during the computation. This tracking means
J correction technology utilise extensive quantum teleportation
that we do not need to perform active quantum corrections
protocols.Theseprotocolsareintrinsicallyprobabilisticandresult
3 incorrectionoperatorsthatoccurasbyproductsofteleportation. because of the probabilistic nature of teleportation operations.
2
These byproduct operators do not need to be corrected in This technique is well known in the quantum information

the quantum hardware itself. Instead, byproduct operators are community and routinely ignored (referred to as working in
]
h trackedthroughthecircuitandoutputresultsreinterpreted. This the Pauli frame). However, to our knowledge, no details on
p tracking is routinely ignored in quantum information as it is the algorithm necessary to perform this tracking have been
- assumed that tracking algorithms will eventually be developed. presented.
t In this work we help fill this gap and present an algorithm for
n
tracking byproduct operators through a quantum computation. The contributions of this paper are as follows. First, we
a
u We formulate this work based on quantum gate sets that are present teleportation-based quantum computing in a generic
q compatible with all major forms of quantum error correction andalgorithmicwayaccessibletothedesigncommunity.There
[ and demonstrate the completeness of the algorithm. are many types of corrections that are combined to define

the Pauli frame of a quantum computation. The two most
1
v I. INTRODUCTION importantarisefromteleportationoperationswhenperforming
arbitrary rotations and the second arises from QEC decoding
2
Quantum computing promises exponential speed-up for
operations.Withoutlossofgeneralitywewillfocusonthefirst.
7
a number of relevant computational problems. Building a
8 By doing this, we completely detach our description from a
scalable and reliable quantum computer is one of the grand
5 particular type of QECC; in fact, an arbitrary QECC can be
challenges of modern science. While small-scale quantum
. applied on top of the basic scheme with a minor adjustment
1 computers are routinely being fabricated and operated in the
to the algorithm.
0
laboratory [1], [2], [3], [4], [5], [6], they can only serve
4 as feasibility studies, and fundamental breakthroughs will be Second, we introduce a new algorithm for Pauli tracking.
1
required before a truly practical quantum computer can be This algorithmallows us to postponecorrectionsuntil the end
:
v built.Asthesizeofcomputerswithinthereachofstate-of-the- ofcomputationwherewe adjusttheoutputofthecomputation
i arttechnologiesincreases,thefocusofinterestshiftsfromtheir based onthe currentstate ofthe Pauliframe.The algorithmis
X
basic physical principles to structured design methodologies completely classical and can be implemented in software and
r thatwillallowto realise large-scalesystems[7], [8],[9], [10]. run on the control computerrather than on the quantum hard-
a
ware. We formalise the algorithm and prove its correctness.
A given technology is suited for construction of general- Experimental results show that Pauli tracking is efficient.
purpose quantum computers if it supports a direct realisation
of a universal quantum gate set which can implement or ap- The paper is organised as follows: In section II we intro-
proximatearbitraryfunctions[11].Moreover,today’squantum ducethebasicsofquantumcomputation.SectionIIIdetailsthe
systems exhibithigh errorrates and require effectivequantum compatiblegatesetsforfault-tolerant,error-correctedquantum
error-correcting codes (QECC) [12]. Consequently, building computation and section IV introduces teleportation-based
a practical quantum computer requires an universal gate set quantumgates. Finally,section V illustrates the Paulitracking
which can be implemented in an error-corrected manner. algorithm and section VI presents several simulation results.
In this paper, we consider a class of quantum circuits
II. QUANTUM COMPUTING
based on an universal gate set that consists of just two
types of operations: injection of specific quantum states into Quantum circuits represent and manipulate information in
the circuit and the controlled-not (CNOT) operation. Using qubits(quantumbits). While classical bits assume either logic
the technique of quantum teleportation, state injections are value 0 or 1, qubits may be in superposition of these two
mapped to rotational gates that together with the CNOT values.A single qubithasa quantumstate ψ =(α0,α1)T =
operationprovideuniversality[20].Theadvantageofthisgate α0 0 + α1 1 . Here, 0 = (1,0)T an|di 1 = (0,1)T
| i | i | i | i
set is that it can be seamlessly integrated into very advanced are quantum analogons of classical logic values 0 and 1,
respectively.α0 andα1 arecomplexnumbercalledamplitudes qubit t. Below are the matrices of the CNOT(c = 1,t = 2)
with α0 2 + α1 2 = 1. 0 and 1 are orthonormal vectors and the H I operations.
and fo|rm|a ba|sis|of C2. | i | i ⊗
1 0 0 0 1 0 1 0
A state (α0,α1)T may be modified by applying single- CNOT=0 1 0 0 H I= 1 0 1 0 1 
qubit quantum gates. Each quantum gate corresponds to a 0 0 0 1 ⊗ √2 1 0 1 0
−
complex unitary matrix, and gate function is given by mul- 0 0 1 0 0 1 0 1
   − 
tiplying that matrix with the quantum state. Two single-qubit
gates that are highly relevant in the context of this paper are A quantum circuit with n qubits and m gates g1,...,gm
the X and the Z gate with the following matrices: takes an input state φ0 C2n and successively applies the
∈
0 1 1 0 transformations corresponding to each gate: φ1 = Mg1φ0;
X =(cid:18)1 0(cid:19) Z =(cid:18)0 −1(cid:19) gφa2te=gMi. Sg2eφle1c;taivnedqsuobfiotrmthe,awsuhreermeeMntgimiasythaels2onb×e2inntemrsaptreirxseodf
withthesegatesandattheendofcomputationtheoutputstate
The application of X to a state results in a bit flip:
X(α0,α1)T = (α1,α0)T; 0 is mapped to 1 , and vice φm of the circuit is completely measured.
| i | i
versa. The application of the Z gate results in a phase flip:
Z(α0,α1)T = (α0, α1)T. Bit and phase flips are used for III. FAULT-TOLERANTGATESET
−
modelling the effects of errors on the quantum state as qubit
It is important to distinguish between the generic mathe-
errorscanbedecomposedintocombinationofbitand/orphase
matical model of quantum computation and subsets of it that
flips. Further important single-qubit quantum gates, in the
are suited for an actual physical realisation. A number of
context of a fully error-corrected system, are
technologies have been suggested for implementing quantum
1 1 1 1 0 1 0 computation [14], [15], [16], [17]. While every unitary com-
H = √2(cid:18)1 1(cid:19) P =(cid:18)0 i(cid:19) T =(cid:18)0 eiπ4(cid:19), plex matrix qualifies as a quantum gate in the mathematical
− formalism, most implementation technologies only allow a
where T2 =P and P2 =Z. direct physical realisation of relatively few gates. Therefore,
universal gate sets, that is, collections of quantum gates that
It is not possible to directly read out the amplitudes of
can representor approximatean arbitraryquantumcircuit, are
a qubits state. A measurement has to be performed instead.
of interest This concept is similar to universal gate libraries
Quantum measurement is defined with respect to a basis and
in digital circuit design, where each Boolean function can be
yields one of the basis vectors with a probability related to
mapped to a circuit composed of, for instance, AND2 gates
the amplitudes of the quantum state. Of importance in this
and inverters. One instance of a universal quantum gate set is
work are Z- and X-measurements.Z-measurementis defined
CNOT,H,P,T . A technologyis suitable for realisation of
with respect to basis (0 , 1 ). Applying a Z-measurementto { }
arbitrary quantumalgorithmsif it has a direct implementation
| i | i
aqubitinstate ψ =α0 0 +α1 1 yields 0 withprobability for at least one universal gate set.
α0 2 and 1 w|ithi proba|biility α|1i2. More|oiver, the state ψ
| | | i | | | i
collapses into the measured state, that is, becomes either 0 A further key requirement for successful realisation of
| i
or 1 . X-measurement is defined with respect to the basis quantum circuits is the ability to perform error correction
(+|,i ), where + = 1 (0 + 1 ) and = 1 (0 during computation. States of actual quantum systems are
| i |−i | i √2 | i | i |−i √2 | i−
1 ). inherently fragile and are affected even by the slightest in-
| i teraction with their environment. Therefore, quantum error-
A multi-qubit circuit processes states represented by an correcting codes (QECC) introduce substantial redundancy to
exponential number of amplitudes. The state of a circuit compensate for impact on the quantum state. For instance,
with n qubits has 2n amplitudes αy with y 0,1 n and Shor’s code [18] uses nine qubits to represent one encoded
y|αy|2 = 1. For example, the state of a 2-q∈ub{it ci}rcuit is qubit:thequbitisfirsttriplicatedinordertodetectandcorrect
Pψ =(α00,α01,α10,α11)T =α00 00 +α01 01 +α10 10 + phase flips, and the resulting qubit triplet is again triplicated
|α1i1 11 . Here, 00 = (1,0,0,0|)Ti, 01 |=i(0,1,0|,0i)T, to protect them against bit flips. It can be shown that this
10|=i(0,0,1,0|)Tiand 11 = (0,0,0,|1)Ti form a basis for constructionissufficientto correctallerrorsaffectinganyone
C| 4.iMeasuringmultipleq|ubiitsofacircuitagainresultsin one of the nine physical qubits. We call the nine qubits used for
basis vector with the probability given by the corresponding encodingphysicalqubitsandone error-correctedqubitlogical
amplitude, α 2. qubit.ThestrengthofaQECCcanbequantifiedbyhowmany
y
| |
physicalqubiterrorsthathavetooccurbeforethelogicalqubit
Quantumgatesmayactonseveralqubitssimultaneously.A
iscorrupted.ForShor’scode,asingleerroronanyofthenine
gatethatactsonn qubitsisrepresentedbya 2n 2n complex
× physical qubit is tolerated; the probability of failure for the
unitarymatrix.Oneimportanttwo-qubitgateisthecontrolled-
Shor’scodeisboundedbytheprobabilityoftwoormoreerrors
notCNOT(c,t)gate,wherethecqubitconditionallyflipsthe
occurringonseparatequbits.Moreadvancedcodescantolerate
stateofthetqubitswhensetto 1 .Moreover,itispossibleto
| i multiple errors and have lower probabilities of failure at the
representa single-qubitgate (or,more generally,a gate acting
expense of more physical qubits to encode a single logical
on less qubits than n) by using tensor product. For example,
qubit.
the22 22matrixH I (whereI istheidentitymatrix)applies
× ⊗
the Hadamard gate H to the first qubit of a two-qubit circuit A fault-tolerant gate for a given QECC acts directly on
whileleavingthesecondqubitunchanged.Thefirstqubitofthe encoded qubits and produces legal encodings with respect
CNOTgateiscalledcontrolqubitcandthesecondisthetarget to that QECC at its outputs. For example, a fault-tolerant
Y R4 φ
implementation of a CNOT gate for the Shor’s code would| i x| i
take 18 physical qubits as inputs, interpret nine of the qubitsφ X
| i •
as the logical control qubit and the other nine qubits as Y R4 φ A R4 R8 φ
the logical target qubit, and produce 18 physical qubits that |φi • Z z| i| i • z z| i
encode the two logical qubits as its outputs. In self-checking | i φ Z
| i
design of classical circuits, error-correcting codes with this
Fig. 1: Teleportation circuits used for (a) R4, (b) R4, (c) R8
propertyare called closed with respectto the gate’soperation; x z z
for example, bi-residue codes are closed under both addition
and multiplication [19]. The closure property is advantageous
because decoding and re-encoding of codewords before and
is , the applied rotation was R ( π/4), i.e., the direction
x
after operation are avoided. This advantage is even more |−i −
of the rotation was wrong. This is easily compensated by
pronounced for quantum circuits with their extremely high
performinganotherrotationbyangleπ/2,namelyapplyingthe
expected error rates. As a consequence, a practical universal
gate R (π/2) = X. It is easily checked that XR ( π/4) =
x x
gateset shouldconsistof fault-tolerantgatesthatallow circuit −
R (π/4). Consequently, quantum teleportation must be fol-
x
operation with errors continuously taking place.
lowed by executing the X gate at the obtained state if the
measurement result is . We call this X-correction. Note
|−i
IV. TELEPORATION-BASEDQUANTUM COMPUTING that the decisionwhetherX-correctionis requiredis based on
classical information(a measurementresult) and can be taken
In this paper, we focus on a set of operations that can be
by a classical computer.
implementedin a fault-tolerantmannerwith respectto several
state-of-the-art QECC [20]. This set consists of the CNOT Gate R4 : The two possible Z-measurement results from
z
gate and two state injection operations. State injection refers the circuit in circuit of Fig. 1b implementing R4 are 0 and
z | i
to initializing a qubit in one of the two following states 1 . The state 0 indicates a correct teleportation where the
| i | i
resulting state is ψ = R (π/4)φ . A measured state 1 is
A = 1 (0 +eiπ/4 1 ) Y = 1 (0 +i1 ). anindicatorforth|esitate ψzf =α|1 i0 iα0 1 wherethe|iniput
| i √2 | i | i | i √2 | i | i state was φ = α0 0 +|α1i1 . In|orid−er to|oibtain the correct
| i | i | i
Using these states and a technique called quantum teleporta- state, a Z operation followed by the X operation is applied,
tion,itispossibletoobtainthefollowingthreequantumgates: as it is easily verifiedthat ψ =XZ ψf =α0 0 +iα1 1 =
R4 φ . This operation is c|allied XZ c|orriection.| i | i
π 1 1 i z| i
Rx(cid:16)4(cid:17)= √2(cid:18)−i −1 (cid:19) GFiagt.e1cR).z8Th:e fiTrshtistelgeaptoeraitsioinmmplaepmsesntateted Ain ttwooanstiangteersm(esdeie-
ate state, which is then given to the R4| giate from Fig. 1b
π 1 0 π 1 0 that also incorporates a teleportation. Tzhe following three
Rz(cid:16)4(cid:17)=P =(cid:18)0 i(cid:19) Rz(cid:16)8(cid:17)=T =(cid:18)0 eiπ8(cid:19) measurement outcomes have to be distinguished:
In the Bloch sphere representation of a quantum state, Rx(θ) 1) If the first measurementresults in 0 , the intermedi-
andRz(θ)standforarotationaroundtheX-andtheZ-axisby ate state is the correct result alrea|dyi. No correction
angleθ;see[11]fordetails.Weusethefollowingabbreviations is required, and the second rotation (including the
Rfozr(πbr/e8v)ity≡: RTx4. :U=sinRgx(thπe/4r)e;lRatiz4on:=shipRzH(π/=4)R≡z4RPx4;RRz4z8, t:h=e caoprprleiespdo.nding measurement) does not have to be
completeuniversalgateset CNOT,H,P,T canbeobtained 2) If the first measurement results in 1 and the in-
{ }
basedonCNOT,stateinjectionandquantumteleportation.All | i
put state was φ = α0 0 + α1 1 , the calculated
cthoemsepugtaatteiosnar[e20c]o,m[p2a1t]ib.leHowwitehveerr,roqr-ucaonrtruemctedte,lefapuolrtt-attoiloenranist state is |ψf1i =| iXRz4†|φ|ii= α0||1ii+e−iπ/4α1|0i.
This state will be used as an input for the π/2
probabilistic itself and may require (classical) correction that correctional rotation (R4 from Fig. 1b). If the sec-
will be tracked. This is described in detail below. z
ond measurement returns 1 , then the correction
| i
The rotational gates R4, R4 and R8 are constructed by succeeded, and no further corrections are necessary:
combining state injection wxith qzuantumzteleportation. Apply- |ψi=α0|0i+eiπ/4α1|1i=Rz8|φi.
ing the three employed rotational gates to an arbitrary state 3) If the first measurement returns 1 , and the second
q|φuibibtyisqiunaitniatulimsedteilnepsotarttaetioYniosrshAow(ndeinpeFnidgi.n1g.oAnnthaeudxeilsliiraerdy pmreoadsuucreemsteantet yψiefld2s |=0i,iαth0e0n t+he|eR−iiz4π/c4oαrr1ec0ti.onThweinll,
rotation), and a CNOT ga|teiis ap|pliied at the qubit that holds as seen above|, theiXZ co|rriection leads|toi ψ =
|φi and the auxilliary qubit (the control and target qubits are XZ|ψf2i=α0|0i+eiπ/4α1|1i=Rx8|φi. | i
denoted by and , respectively). Finally, a measurement
• ⊕ Insummary,teleporationsareprobabilisticandeitherX or
(eitherX orZ)isperformedatthecontroloutputoftheCNOT
XZ corrections may be required depending on the outcomes
gate, indicated in Fig. 1 by an encircled X or Z. The effect
ofthemeasurement.Itisimportanttounderstandthatthisnon-
at the target output is shown in Fig. 1
determinismisnotduetoerrorsbutisinherenttoteleportation-
Gate R4 : The X-measurement in circuit of Fig. 1a yields based quantum computing. Algorithm 1 summarises the com-
x
either + or . If the measurement result is + , then the plete computation procedure incorporating all the required
| i |−i | i
desired rotation R (π/4) was executed and the new state at corrections in detail. The algorithm assumes a circuit that has
x
the output of the circuit is correct. If the measurement result already been mapped to the universal gates set consisting of
Algorithm 1 Teleportation-based quantum computation and the classical control computer. This may have a detri-
mental impact on the speed of computation and is ultimately
Input: n-qubit quantum circuit with m gates g1,...,gm
{CNOT,Rx4,Rz4,Rx8}, input state φ0 ∈ uconrnreeccetisosnarsyc.aInnbtehpisosstepcotnioend,towtehedeenmdoonfstcraatlecuhlaotwionawppitlhyoinugt
Output: Output state φ
m
losingaccuracy.Forthispurpose,teleportation-basedquantum
1: for i:=1 to m do
computation method is modified as follows.
2: if gi is a CNOT gate then
3: // Apply CNOT to current state
TheconsideredcircuitsstillconsistofCNOTgatesandthe
4: φi :=Mgφi+1; three types of rotational gates implemented by state injection
5: else if gi is a Rx4 gate on qubit k then and quantum teleportation. Measurements are still performed
6: Introduce new qubit l; inject state Y on l;
| i during quantum teleportation, however their outcomes are
7: Perform CNOT(k, l); X-measurement on qubit k;
stored in a variable rather than used for immediate correction.
8: if measurement result is + then
9: apply X-correction on|quibit l; mFoeraseuarcehmreontat.tiNonoatlegtahtaetgbi, varia+ble,bi holidfsgtheisreasuRlt4ogfatthee,
10: end if b 0 , 1 if g is a Ri4∈ga{te|, ib |−i}00 ,i01 , 10x, 11
1121:: elsReeipflgacieisqaubRitz4kgianteφtih−e1nby qubit l to obtain φi; iofifg∈tiwi{os|caioRn|sz8eicg}uattiev,ewimheeraesupraezimrseonftsv.ailu∈es{r|efeir t|o tihe|ouitc|omie}s
13: Introduce new qubit l; inject state Y on l;
| i
14: Perform CNOT (l, k); Z-measurementon k; We derive an algorithm that calculates, for a given combi-
15: if measurement result is 0 then nation of b values, the vector of equivalent output correction
| i i
16: Apply XZ-correction on qubit l; statusesS =(s1,...,sn).Forqubitk,sk assumesoneoffour
17: end if valuesthatindicatetherequiredcorrections:I (nocorrection),
18: Replace qubit k in φi 1 by qubit l to obtain φi; X (X-correction, i.e., a bit flip), Z (Z-correction, or a phase
19: else if gi is a Rz8 gate th−en flip),andXZ (bothX-andZ-correction).ThevaluesinS are
20: Introduce new qubit l; inject state A on l; calculated such that running the teleportation-based quantum
| i
21: Perform CNOT(l, k); Z-measurement on k; computing(Algorithm1)withoutapplyingcorrectionsinLines
22: if measurement result is 0 then 8–10, 15–17 and 22–31 and applying the correction in S to
| i
23: Introduce new qubit l′; inject state Y on l; the obtained output state is equivalent to teleportation-based
| i
24: Perform CNOT(l′,l); Z-measurement on l; quantum computing with immediate correction.
25: if measurement result is 0 then
| i
26: Apply XZ-correction on qubit l′; S is calculated by propagating (tracking) the correction
27: end if status (s1,...,sn) through the circuit. The calculation is
28: Replace qubit k in φi 1 by qubit l′ to obtain φi; applied after the teleportation-based quantum computation
29: else − took place and the measurement results b associated with
i
30: Replace qubit k in φi 1 by qubit l to obtain φi; all rotational gates gi are available. Each sk is initialised
31: end if − to I (no correction). Then, the gates are considered in their
32: end if regular order g1,...,gm. If gi is a rotational gate on qubit k,
33: end for its b is consulted to decide whether a correction is needed
i
34: return φm; and the sk is updated (the correction status is propagated).
The propagated correction status shows up at the inputs of
subsequentrotationaland CNOT gatesandmust be taken into
theCNOTgateandthethreeconsideredrotationalgates.Note accountwhencalculatingthe correctionstatusat theoutputof
thattheonlyoperationsappliedarestateinjectionsandCNOT thatgates.Weintroducethecorrectionstatustrackingfunction
gates, and that these operations are compatible with standard τ that formalises the propagation.
fault-tolerant error correction.
There are two versions of τ: one for CNOT gates and
The computation contiunuously requests new qubits and for rotational gates. CNOT gates do not employ teleportation
abandons the old ones, such that the total number of used and thereforerequireno corrections;however,correctionsthat
logicalqubitsisnorn+1atanygiventime.Inmanyrelevant originated from rotational gates may show up at the inputs
implementation technologies, hardware for each abandoned of the CNOT gate and have to be propagated to its outputs.
qubitcanbereusedforthenewlyrequestedones.Forexample, Let c and t be the control and the target qubit of the CNOT
ifanewqubitisintroducedforinjectingthe Y state inorder gate,andletsin andsin bethecorrectionstatusesattheinputs
to implement the R4 gate, the old qubit is n|o lionger required of these qubitsc, respecttively. Then, τ(sin,sin) producesa pair
x c t
aftertheX-measurement,anditcanbeusedforimplementing of correction statuses (sout,sout) at the outputs of the CNOT
c t
further rotational gates. gates by the following calculation:
V. PAULI TRACKING ALGORITHM socut = (cid:26) ssiicnn Z iiff ssiitnn ∈{ZI,,XX}Z (1)
Teleportation-based quantum computing suffers from the c ⊕ t ∈{ }
nmeecaessusriteymetontcroensudlittsio.nAapllpylypinergfocromrreccotirorencitnionresalbtaismede,oimnmthee- sotut = (cid:26) ssitinn X iiff ssicinn ∈{IX,,ZX}Z (2)
t ⊕ c ∈{ }
diately after the rotational operation, such as in Algorithm 1,
results in significant interaction between quantum hardware Here, s Z and s X are flippingthe status ofthe respective
⊕ ⊕
i 1 2 3 4 5 6
TABLE I: Correction status tracking τ for rotational gates
g R4 CNOT R8 CNOT R4 R4
i x z z z
gi sikn bi sokut gi sikn bi sokut bi + n/a 10 n/a 0 1
I |+i XI I |010∗i XIZ s1 I |Ii I |Ii Z |Zi |Xi
|−+i X |11i I s2 I I I XZ XZ X X
R4x Z ||−+ii XI Z ||010∗ii XZ Initially, s1 and s2 are set to I (no correction required).
XXZ |||−+iii ZZI R8z X |||100101iii XZIZ qSuinirceedba1nd=s1|+reimisaimnseIas.uTrheedCfoNrOgTatgeagte1,g2nodoceosrrneocttiionntroisdurcee-
|−0i I |1 i I new corrections. The measurementoutcomeb3 = 10 of gate
I ||10ii XZZ ||0001∗ii XZ g3 necessitates the XZ-correction, as can be seen|iniTable I.
R4z Z ||10ii XXZ XZ ||1101ii XZ SZi,nctehethceotrarregcetitoinnpsutatt(uqsubaittt2h)eofcothnetroClNqOuTbitgaitsedge4teirnmcliundeeds,
XXZ |||101iii XXI | i uunsicnhganEgqe.d2.,ba5s=s10=sle1a⊕dsZto=s2I ⊕=ZX=anZd,bw6h=ile1s2 lreeamdasintos
| i s1 = X. As a res|ulit, X-corrections must be appl|ieid at both
outputs of the circuit. (cid:3)
Algorithm 2 Pauli tracking
Input: n-qubit quantum circuit with m gates g1,...,gm The correctness of the tracking algorithm is now formally
CNOT,R4,R4,R8 , measurement results b for ever∈y proven. The following two lemmas formulate the validity of
r{otational gaxte gz x} i the τ function for the individual gates. They can be verified
i
Output: Equivalentoutputcorrectionstatus S =(s1,...,sn) foreverycombinationofinputsforτ.Insteadofprovidingthe
completeproof,wequotethecalculationforonespecificinput
combination, whereas the derivations for other combinations
1: s1 :=s2 := :=sn :=I;
2: for i:=1 to··m· do are similar.
3: ifgi isaCNOTgatewithcontrol/targetqubitsc/tthen Lemma 1 Let g be a CNOT gate with correction status sin
4: (sc,st):=τ(sc,st); // Use Eqs. 1, 2 at its control andi sin at its target input and (sout,sout) =c
5: else if gi is a rotational gate on qubit k then τ(sin,sin). Then, petrforming the sin-correction atcthetcontrol
6: sk :=τ(sk,bi); // Use Table I inpuct antd the sin-correction at thec target input followed by
7: end if application of CtNOT is the same function as applying the
8: end for CNOT gate first and performingsout-correctionat the control
9: return S =(s1,...,sn); output and the sout-correction at tche target output.
t
Proof for sin = X,sin = I: According to Eqs. 1, 2,
c t
(sout,sout) = τ(X,I) = (X,X). Without loss of generality,
correction in s: asscumetthat the control and target qubit of the CNOT gate
are qubit 1 and 2 respectively. Then, the X-correction at
s s Z s X s s Z s X
⊕ ⊕ ⊕ ⊕ the control qubit is described by matrix X1 = X I and
I Z X X XZ I ⊗
the X-correction at the target qubit is described by matrix
Z I XZ XZ X Z
X2 =I X. Performing the correctionsfirst followed by the
⊗
CNOT operation corresponds to the matrix CNOT X1 I
· ·
while the CNOT operation followed by the two corrections
For a rotational gate g on qubit k, the τ function takes
i are described by the matrix X1 X2 CNOT.
the pre-stored measurement result bi and the correction status · ·
sin at its input and calculates the correction status sout at The equivalence of these matrices is shown below:
k k
its output. The values calculated by τ for the three types of
rotational gates considered are given in Table I. CNOT X1
·
1 0 0 0 1 0 0 0 0 0 1 0
Algorithm 2 summarises the Pauli tracking procedure. 0 1 0 0 0 1 0 0 0 0 0 1
=  = 
0 0 0 1 · 0 0 0 1 0 1 0 0
Example: Consider the two-qubit circuit in Fig. 2. Assume  0 0 1 0  0 0 1 0   1 0 0 0 
that teleportation-based quantum computing has been done     
 0 0 1 0  0 1 0 0  1 0 0 0 
without applying corrections. The recorded measurement re-
0 0 0 1 1 0 0 0 0 1 0 0
sults bi and the calculated correctionstatuses S =(s1,s2) are = 1 0 0 0 · 0 0 0 1 · 0 0 0 1 
shown in the following table.    
 0 1 0 0  0 0 1 0  0 0 1 0 
   
=X1 X2 CNOT  
· ·
1 2 3 4 5 6
0 R4 R4 All other combinations can be calculated similarly. (cid:3)
x • • z
1 R8 R4 Lemma 2 Let g be a rotationalgate, sin the correctionstatus
z z
at its input, b the outcome of the associated measurementand
Fig.2:Quantumcircuitimplementedusingfault-tolerantgates sout = τ(sin,b) the tracked correction status at its output
according to Table I. Then, performing the sin correction,
VII. CONCLUSIONS
TABLE II: Run-times RT (in seconds) of the Pauli tracking
Wehavepresentedanalgorithmthatcanbeusedtoperform
algorithm for circuits with n qubits and m quantum gates
Pauli tracking on quantum circuits compatible with all major
n m RT [s] n m RT [s] n m RT [s] classes of QECC. This result helps fill an important gap in
100 1000 0 1100 1000 0.011 5100 1000 0.052 the classical control software needed for large-scale quantum
100 5000 0.002 1100 5000 0.069 5100 5000 0.309
computation.Pauli tracking is instrumentalfor both error cor-
100 10000 0.004 1100 10000 0.128 5100 10000 0.709
100 20000 0.011 1100 20000 0.300 5100 20000 1.930 rectionandforteleportationbasedprotocolsandthisalgorithm
100 50000 0.030 1100 50000 0.680 5100 50000 5.435 is easily adjustable to incorporate the required tracking for a
specific implementation of quantum error correction. Future
work will be focused on adapting this algorithm to popular
applying g and perforimng,if needed, correction accordingto
error correction techniques such as Topological codes [13],
b,yieldstheequivalentstateasapplyingg firstandperforming
[22] which requires even more intensive Pauli tracking due
sout-correction.
to the enormous number of teleportation gates necessary to
Prooffor gate R4, b= 1 und sin =Z: Since b= 1 , regu- perform a fully fault-tolerant, error corrected computation.
z | i | i
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