Compiling quantum algorithms for architectures with multi-qubit gates

0.68 MB·

by  Esteban A. Martinez

Compiling quantum algorithms for architectures with multi-qubit gates
Esteban A. Martinez,1 Thomas Monz,1 Daniel Nigg,1 Philipp Schindler,1 and Rainer Blatt1,2
1Institut fu¨r Experimentalphysik, Universita¨t Innsbruck, Technikerstraße 25/4, 6020 Innsbruck, Austria
2Institut fu¨r Quantenoptik und Quanteninformation,
O¨sterreichische Akademie der Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria
Inrecentyears,small-scalequantuminformationprocessorshavebeenrealizedinmultiplephysical
architectures. Thesesystemsprovideauniversalsetofgatesthatallowonetoimplementanygiven
unitary operation. The decomposition of a particular algorithm into a sequence of these available
gates is not unique. Thus, the fidelity of the implementation of an algorithm can be increased by
choosing an optimized decomposition into available gates. Here, we present a method to find such
a decomposition, taking as an example a small-scale ion trap quantum information processor. We
demonstrate a numerical optimization protocol that minimizes the number of required multi-qubit
entanglinggatesbydesign. Furthermore,weadaptthemethodforstatepreparation,andquantum
6
algorithms including in-sequence measurements.
1
0
2
CONTENTS available set of gates is known as universal if it is possi-

l bleto findsucha decompositionfor anarbitraryunitary
u
quantum operation acting on the qubit register.
J I. Introduction 1
Acanonicaluniversalsetofgatesconsistsoftwo-qubit
1 A. Experimental toolbox 2
CNOT gates and arbitrary single qubit rotations. There
2
II. Compilation of local unitaries 2 exist deterministic algorithms that provide near-optimal
] decompositions of unitaries in terms of these gates [6].
h
However,thesetofgatesthatyieldsthehighestfidelities
p III. Compilation of general unitaries 4
depends on the particular experimental implementation.
- A. Compilation in layers 4
t In particular, two-qubit CNOT gates may not be the
n B. Numerical optimization 5
most efficient to implement. Architectures like trapped
a
u C. Compilation of isometries 6 ions [7, 8] or atom lattices [9] include in their toolboxes
q D. Compensation of systematic errors 7 high-fidelitymulti-qubitgatesthatactontheentirequbit
[ register (see section IA). Implementing two-qubit gates

IV. Conclusions and outlook 7 in terms of these requires refocusing [10] or decoupling
2
v [7] techniques, and thus increases the overhead. There-
9 V. Acknowledgements 8 fore it is desirable to find a direct decomposition of the
1 target unitary into the available operations. In general,
8 A. Compiling local unitaries 8 the number of multi-qubit gates needs to be minimized,
6
1. Finding basis changes 8 since these are more prone to errors than local gates.
0
2. Writing a unitary as a product of two Compilingunitariesusingmulti-qubitgatesthatacton
.
1 equatorial rotations 9 the whole qubit register is more challenging than using
0
3. Unitaries up to a collective Z rotation 9 two-qubitgates. Evenifasequenceimplementscorrectly
6
aunitaryforN qubits,itmightnotworkforN+1qubits,
1 4. Unitaries up to independent Z rotations 9
since additional “spectator” qubits will also be affected
:
v by the sequence instead of being left unchanged [11].
References 11
i
X Therefore, one has to define a qubit register of interest
where the unitary will be compiled, and the experimen-
r
a tal implementation of the resulting sequence has to be
I. INTRODUCTION limited to this subregister, as explained in Section IA.
Moreover,theexistinganalyticalmethodsforfindingde-
Quantum technologies open new possibilities that are compositionsofunitariesintermsoftwo-qubitgates(see
inaccessible with current classical devices, ranging from forinstanceRef.[12])donotseemtoapplytomulti-qubit
cryptography[1,2]toefficientsimulationofphysicalsys- gates. Therefore, in this work we employ an approach
tems [3–5]. To utilize the full computational power of based on numerical optimization.
quantum systems, one needs a universal quantum com- A similar algorithm for finding multi-qubit gate de-
puter: a device able to implement arbitrary unitary op- compositionshasbeenstudiedinRef.[11],whereoptimal
erations, or at least to approximate them to arbitrary controltechniquesareusedtofindapulsesequencefora
accuracy. However, in any specific physical system, only given target unitary operation. The procedure described
acertainsetofoperationsisreadilyavailable. Therefore, therestartswithlongsequencesandthenremovespulses,
itisnecessarytodecomposethedesiredunitaryoperation ifpossible. Thisoftenresultsinsequenceswithmoreen-
asasequenceoftheseexperimentallyavailablegates. An tangling operations than actually required. In this work
2
we present an algorithm designed to produce decompo- • Entangling Mølmer-Sørensen (MS) gates [15], with
sitions with a minimal number of entangling gates. In arbitrary rotation angle and phase MS (θ). Here θ
φ
addition,weintroduceadeterministicalgorithmforfind- istherotationangleandφisthephaseofthegate,
ingdecompositionsoflocalunitaries. Wealsoextendthe resulting in:
algorithmtooperationsrequiredforstatepreparationor
measurement, which are particular cases of more general MS (θ)=e−iθ(Sxcosφ+Sysinφ)2/4, (5)
φ
operations known as isometries [13, 14].
Thepaperisorganizedasfollows: inSectionIAwede- where Sx,y = σ1x,y +···+σNx,y are the total spin
scribepreciselywhichgateswewillconsideraspartofour projectionsonthex ory axes,asbefore. Forφ=0
experimentally available toolbox, and review some ar- or φ=π/2 we obtain gates that act around the X
chitectures for quantum information processing to which or Y axes, which we will denote:
the methods described in this work can be applied. In
Section II we show an analytic algorithm to compile lo- MSx,y(θ)=e−iθS2x,y/4. (6)
cal unitaries, which can be used to find efficient imple-
As mentioned before, it is desirable to be able to
mentations of state and process tomographies. Finally,
restrict the action of the MS gate to a particular
in Section III we describe and analyze an algorithm to
qubit subset. This can be done experimentally by
compile fully general unitaries which relies on numerical
spectroscopically decoupling the rest of the qubits
optimization.
from the computation [7], or by addressing the MS
gate only on the relevant subset of the qubits [16].
A. Experimental toolbox
This set of gates, or equivalent ones, are available in
several trapped-ion experiments [7, 17, 18]. A similar
Several quantum information processing experiments toolbox of operations is available for architectures based
based on atomic and molecular systems have similar on trapped-ion hyperfine qubits. For example, Ref. [8]
toolsets of quantum operations at their disposal. Of- describes high-fidelity microwave gates applied to a sin-
ten, it is convenient to apply collective rotations on an glehyperfine43Ca+qubit. Inamulti-qubitsystem,these
entire qubit register. These collective (yet local) gates, gates would drive collective rotations like the ones de-
combined with addressed operations (typically rotations scribed above. In addition, Ref. [19] describes a Raman-
aroundtheZaxis)allowonetoimplementarbitrarylocal drivenσz⊗σz phasegateontwoqubits,whichappliedto
unitaries,asweshowinSectionII.Togetherwithsuitable a many-qubit register would act analogously to the MS
multi-qubit operations, arbitrary quantum unitaries can gate already described.
be implemented. In this work we consider the following Recently, an implementation of high-fidelity gates in a
set of gates: 2D array of neutral atom qubits was reported [9]. The
toolbox described there consists of global microwave-
• Collective rotations of the whole qubit register driven gates and single-site Stark shifts on the atoms,
about any axis on the equator of the Bloch sphere which are completely equivalent to the local operations
C(θ,φ). Here θ is the rotation angle and φ is the described before for the trapped ion architecture. A
phase, so that: multi-qubitCNOTgate,equivalenttotheMSgate,could
also be implemented by means of long-range Rydberg
C(θ,φ)=e−iθ(Sxcosφ+Sysinφ)/2, (1) blockade interactions [20].
where S = σx,y +···+σx,y are the total spin
x,y 1 N
projections on the x or y axes, and σjx,y,z are the II. COMPILATION OF LOCAL UNITARIES
respective Pauli operators corresponding to qubit
j. For the sake of brevity we also define rotations
Local unitaries can be written as a product of single-
around the X and Y axes as:
qubit unitaries. In this section we show a fully deter-
ministic algorithm that produces decompositions of any
X(θ)=C(θ,0), (2)
local unitary as a sequence of collective equatorial rota-
Y(θ)=C(θ,π/2). (3) tions and addressed Z rotations, as described in section
IA. The decompositions presented here are optimal in
• Single qubit rotations around the Z axis Z (θ), the number of pulses. These techniques are particularly
n
where θ is the rotation angle, and n is the qubit useful for the implementation of state and process tomo-
index: graphies, as exemplified in figure 1, since both require
only local operations at the beginning and end of the
Zn(θ)=e−iθσnz/2, (4) algorithm.
Let us consider a register of N qubits, and a local uni-
with σz being the Pauli Z operator applied to the taryU =U ⊗U ⊗···⊗U tobeappliedtothem,where
n 1 2 N
n-th ion. U is the action of the unitary on the i-th qubit. If the
i
3
togetheraccordingtowhichofthemexperiencethesame
single-qubit unitary U . For an application such as state
i
tomography, thereareonlythreepossibleunitariestobe
appliedtoeachqubitintheregister(showninFigure1),
sinceoneonlywantstoperformameasurementinoneof
three different bases. Therefore, effectively we only need
to consider three qubits, in which case trying out all the
permutations is perfectly feasible.
We will describe now how to compile a generic local
FIG.1. Pulsesequencetoperformaprojectivemeasurement unitary U = U1 ⊗ U2 ⊗ ··· ⊗ UN exactly, using a de-
on the {X,Y,Z} bases for qubits {1,2,3}, respectively. composition of the form (8). Let us first note that the
unitaries in Eq. (8) act on the N-qubit Hilbert space,
which is the tensor product of the single-qubit Hilbert
sameoperationhastobeappliedtomorethanonequbit spaces. For the sake of simplifying the notation, we will
(U = U ), we can replace both with a single instance now refer to these unitaries as C˜ , Z˜ , and will reuse the
i j i i
of the operation, and then apply the same addressed ro- notations C and Z for their action on the single qubits,
i i
tations on all qubits subject to the same operation U . so that:
i
Therefore, weonlyhavetoconsiderthecasewhereevery
Ui is unique. C˜i =Ci⊗Ci⊗···⊗Ci, (9)
In order to apply a general local unitary to each qubit Z˜ =1⊗1⊗···⊗Z ⊗···⊗1,
i i
we need to have at least three degrees of freedom per
qubit [6], so the decomposition must have at least 3N where 1 is the 2×2 identity matrix, and Z appears at
i
free parameters. During the sequence at least N − 1 the i-th place (since it only addresses the i-th qubit).
of the qubits must eventually be addressed, since a dif- In terms of these single-qubit unitaries, factoring
ferent unitary has to be applied to each qubit. There- Eq. (8) for each qubit we obtain N equations:
fore, a sequence of addressed operations of the form
Z (θ ),Z (θ ),...Z (θ ) must be included in the U =C(cid:48) C ···C Z C , (10)
1 1 2 2 N−1 N−1 1 N N 2 1 1
decomposition. These provide N − 1 parameters, so U =C(cid:48) C ···Z C C ,
2N +1 more degrees of freedom are required. The most 2 N N 2 2 1
.
economic way to provide these is by means of collective .
.
gates C(θ ,φ ), which have two degrees of freedom each,
i i U =C(cid:48) C ···C C .
so the shortest sequence possible must include at least N N N 2 1
N global operations, for a total of 3N −1 free param-
From the last equation we can determine C(cid:48) C :
eters. One additional degree of freedom remains, so we N N
must add a last gate. This can be either an addressed
C(cid:48) C =U C−1C−1···C−1 , (11)
operation on qubit N or a collective gate. If we add an N N N 1 2 N−1
addressed gate Z , we obtain a sequence of the form:
N andeliminatingthisfactorfromtheremainingequations
we obtain:
U =Z C Z C ···Z C Z C , (7)
N N N−1 N−1 2 2 1 1
U−1U =C−1Z C , (12)
where C = C(θ ,φ ) and Z = Z (θ ) are collective and N 1 1 1 1
i i i i i i
single-qubit rotations respectively, as explained in Sec- UN−1U2 =C1−1C2−1Z2C2C1,
tion IA. Such a sequence is useful for compiling local .
.
unitaries up to arbitrary phases, as explained in Appen- .
dices A3 and A4. The second alternative is to add a U−1U =C−1C−1···C−1 Z C ···C C .
collective rotation C(cid:48) : N N−1 1 2 N−1 N−1 N−1 2 1
N
We solve each equation in (12) consecutively. To solve
U =CN(cid:48) CNZN−1CN−1···Z2C2Z1C1, (8) the first equation in (12), let us notice that its left-hand
side is a known unitary, which can be written as:
which is the type of sequence we consider in this section.
For particular unitaries, some of the Ci and Zi in UN−1U1 =e−iα1u1/2, (13)
Eq. (8) may actually be the identity, in which case the
sequence is simpler. Since the decomposition depends whereα istheangleoftherotationandu itsgenerator.
1 1
on the ordering of the qubits, by reordering them a sim- The right-hand side is simply a rotation around Z and a
pler sequence might be obtained. For small numbers of changeofbasis. Therefore,therotationangleofZ must
1
qubits, one can compile the unitary for every possible be equal to α , and the change of basis must be such
1
permutation, although this becomes inefficient for large that:
numbers of qubits. However, let us remember that, for
the purposes of the compilation, the qubits are grouped u =C−1σ C . (14)
1 1 z 1
4
We show in Appendix A1 how to find the generator and We seek decompositions directly in terms of multi-
angle of the collective rotation C . qubit entangling gates, since these are often more effi-
1
Having determined C , we can write the second equa- cient than decompositions in terms of two-qubit gates.
1
tion in (12) as: For example, a Toffoli gate can be implemented using
only 3 Mølmer-Sørensen (MS) gates [11], while 6 CNOT
C U−1U C−1 =C−1Z C . (15) gates are needed to implement it [21]), and a Fredkin
1 N 2 1 2 2 2
gate can be implemented using 4 MS gates [22], while
As before, the left-hand side of this equation is a known theleastnumberoftwo-qubitgatesrequiredis5[23]. As
unitary, and the right-hand side consists of a rotation described in section IA, many equivalent types of entan-
around Z and a change of basis, so the rotation angle glinggatesareexperimentallyavailable. Wewillconsider
θ and generator of the change of basis C can be found MS gates, but the methods shown here are applicable to
2 2
as for the previous equation. This procedure can be re- any entangling gate that forms a universal set together
peated until all of the C and Z with k ≤ N −1 are with local operations.
k k
determined. The last collective operations C and C(cid:48)
N N
can be determined from equation (11). For this we need
to decompose an arbitrary unitary into a product of two A. Compilation in layers
equatorialrotations; thiscanbedoneasexplainedinap-
pendix A2. In many quantum information processing experiments
We have shown so far how to compile a local unitary the most costly operations in terms of fidelity are entan-
exactly. However, in certain cases the constraints on the gling gates. Therefore, when trying to compile a unitary
targetunitaryareweaker, sothatitcanbeimplemented we seek to minimize the number of those. A straight-
with a simpler sequence. For instance, a unitary that forward way to do this is to use pulse sequences where
is followed by global gates whose phase can be freely ad- layers of local unitaries and entangling gates are applied
justedmustonlybespecifieduptoacollectiveZrotation consecutively, as shown in figure 3.
afterwards, since this rotation can be absorbed into the
phase. This removes one free parameter from the se-
quence, thus simplifying its implementation. The details
of this procedure are presented in appendix A3. An-
other case of interest is when the target unitary is speci-
fied up to arbitrary independent Z rotations afterwards,
for instance when the unitary is followed by a projective
measurement on the Z basis. This is particularly use-
ful for tomographic measurements; details are shown in
Appendix A4.
III. COMPILATION OF GENERAL UNITARIES FIG. 3. Sequence with layers of local and entangling gates
applied consecutively.
In section II we studied how to compile local unitaries
Any unitary can be decomposed in terms of single-
in terms of collective and addressed rotations. However,
qubit gates and two-qubit CNOT gates [24]:
a universal quantum computer also requires entangling
unitaries, which must be compiled into the experimen-
U =L CNOT ··· L CNOT L , (16)
tally available local and entangling gates. For example, M M 1 1 0
infigure2weshowadecompositionofaToffoligateinto
whereL denotesanarbitrarylocalunitaryonthewhole
a sequence of local and entangling gates applied consec- i
qubit register and CNOT denotes a gate between some
utively. In this section, we present an algorithm to find i
twoqubits. Atwo-qubitCNOTgatecanbeimplemented
such decompositions for arbitrary unitaries.
inanarbitraryN-qubitregisterasasequenceoflocaluni-
taries and MS (π/8) gates [11]. Therefore, the following
x
decomposition is always possible:
U =L MS (π/8) ··· L MS (π/8) L . (17)
M x 1 x 0
However, some of the local unitaries L in a decomposi-
i
tion of the form (17) may actually be identity, so after
removing them the resulting sequence has the following
FIG.2. DecompositionofaToffoligateintoapulsesequence structure:
of collective equatorial rotations, addressed Z rotations and
entangling Mølmer-Sørensen (MS) gates.
U =L MS (k π/8) ··· L MS (k π/8) L , (18)
M x M 1 x 1 0
5
where the k are integers, and the number M of entan- B. Numerical optimization
i
gling gates is the same or less than in Eq. (17). It is
enough to consider 0 ≤ k ≤ 7, since MS (π) is either
x We have described a general form of a sequence of lo-
the identity for an odd number of qubits, or a π rotation
cal operations and global entangling gates that imple-
around X for an even number of qubits.
ments any desired target unitary. It remains to find the
We now seek to further simplify sequence (18). Every
actual sequence parameters, that is, the rotation angles
single-qubit unitary U on qubit i can be written as a
i and phases of the gates. However, we do not know a pri-
composition of rotations around two different fixed axes
orihowmanyentanglinggateswillbeneededforagiven
[6], which means that we can always choose α , α and
i1 i2 unitary. Therefore we suggest the following algorithm:
α such that:
i3
1. Propose a sequence with M =0 entangling gates.
U =X (α )Z (α )X (α ). (19)
i i i3 i i2 i i1
2. Search numerically for the sequence parameters
Any local unitary L = (cid:81)Ni=1Ui can therefore be written that maximize the fidelity with the target unitary.
as:
3. Ifthesequencehasconvergedtothedesiredunitary
N
(cid:89) (i.e. thefidelityequals1),stop. Otherwiseincrease
L= X (α )Z (α )X (α ), (20)
i i3 i i2 i i1 M by 1 and go back to step 2.
i=1
wheretheproductgoesovertheN qubitsintheregister. Whenperformingthenumericaloptimizationinstep2
Since unitaries acting on different qubits commute, we there might be a number of local optima in addition to
can write this as: thetrueglobaloptimum,makingfullydeterministicopti-
mizationmethodsdifficulttoapply. Weapplythereforea
(cid:89)N (cid:89)N (cid:89)N repeated local search, where an efficient deterministic op-
L= X (α ) Z (α ) X (α ) (21)
i i3 i i2 i i1 timization method is iterated, each time using randomly
i=1 i=1 i=1 determined initial conditions. The initial conditions are
=X˜(cid:48)Z˜X˜, (22) chosen randomly for every optimization run, as experi-
encehasshownusthatstartingclosetopreviouslyfound
where X˜ and Z˜ denote arbitrary products of rotations localminimadoesnotofferanyimprovement. Thesearch
around the X or Z axes for all qubits. Therefore, the is finalized whenever the fidelity with the target unitary
sequence in (18) can be written as: isabovesomepredefinedthreshold, orwhenamaximum
numberoftriesisexceeded. Anadvantageofthismethod
U =X˜M(cid:48) Z˜MX˜MMSx(kMπ/8)···× (23) is that, since each optimization run starts from random
×X˜(cid:48)Z˜ X˜ MS (k π/8)X˜(cid:48)Z˜ X˜ , initial conditions, these are easy to perform in parallel.
1 1 1 x 1 0 0 0
The algorithm chosen for each numerical optimization
and commuting the X rotations with the MS gates we is the quasi-Newton method of Broyden, Fletcher, Gold-
obtain a sequence of the form: farb, and Shanno (BFGS) [25]. The function to be max-
imized is the fidelity of the unitary resulting from the
U =X˜M(cid:48) Z˜MX˜MMSx(kMπ/8)···× (24) pulse sequence with the target unitary. Since we are
×X˜(cid:48)Z˜ X˜ MS (α )Z˜ MS (k π/8)X˜(cid:48)Z˜ X˜ . interested in exact solutions, we reject those solutions
2 2 2 x 2 1 x 1 0 0 0
whose fidelity (normalized to the maximum value pos-
Every odd local unitary (except for the last one) is a sible) is not equal to 1 within some tolerance threshold
product of Z rotations on all qubits, and the even local (usually 1%). The gradient of the fidelity can be calcu-
unitaries can be grouped as L = X˜(cid:48)Z˜ X˜ . Moreover, lated analytically as a function of the sequence param-
i i i i
a collective Z rotation can be extracted from each even eters, which speeds up the computation as compared to
local unitary L and absorbed into the phase of the sub- using several evaluations of the fidelity function.
i
sequent MS gates and collective operations to simplify A previously used approach to this optimization prob-
theimplementationofL . Thereforethesequencecanbe lemwasacombinationoflocalgradientdescentandsim-
i
written as: ulatedannealing(SA)[11],whichalsohelpstoavoidlocal
maxima. However, this method did not make use of the
U =LMMSφM(kMπ/8)···× (25) analyticexpressionforthefidelitygradient,whichspeeds
×L MS (k π/8)Z˜ MS (k π/8)L . up the search. Moreover, its performance depends on
2 φ2 2 1 φ1 1 0 the “topography” of the optimization space and requires
We have thus shown that any N-qubit unitary U can manual tuning of the search parameters to achieve opti-
bedecomposedintoasequenceoftheformshownin(25). malresults. WehavecomparedtheBFGSandsimulated
These sequences always have the same structure, which annealingapproachesbycompiling100randomunitaries
makes it easier to identify patterns if one wants to com- uniformly distributed in the Haar measure as explained
pile families of unitaries, i.e. unitaries that depend on in [26] for different numbers of qubits. In our experi-
some tunable parameter. ence, we find that the BFGS method scales better with
6
the number of qubits than simulated annealing (see Fig- A particularly interesting group of unitaries are Clif-
ure 4). The median number of search repetitions needed fordgates, whichfindapplicationsinquantumerrorcor-
to find the global optimum was 1 in all the cases. rection [29], randomized benchmarking [30], and state
distillationprotocols[6]. Toexplorethedifficultyofcom-
pilingsuchgates,wehavetestedouralgorithmwithran-
domlygeneratedCliffordgates, asexplainedinRef.[31].
3
10
We show in Figure 5 the distribution of the optimal
numberofentanglinggatesrequiredforcompligingtwo-,
2 three- and four-qubit unitaries. Our results agree with
10
the literature [32] for the two-qubit case, since MS gates
arethenequivalenttocontrolled-Z(orCNOT)gates. For
s)
e (101 larger numbers of qubits, the performance of our algo-
m rithm in terms of number of multi-qubit gates required
Ti
is also similar to that of algorithms based on two-qubit
0 gates [32].
10
BFGS
SA
-1
10
1 2 3 4
Number of qubits
FIG. 4. Average time required to find the global optimum
for 100 unitaries randomly distributed in the Haar measure
with the BFGS and simulated annealing methods, using an
Intel©Core i5-4670s CPU 550 @ 3.10 GHz x 4 (one process-
ing thread per optimization run). No data was obtained for
the simulated annealing approach for 4 qubits owing to the
excessive time required.
The exponential scaling of the optimization problem
complexity depends on the number of entangling gates
required to compile a given unitary, which is an intrin-
sic property of the unitary and does not depend on the
searchalgorithm. Itisalreadyknownthatitisnotpossi-
bletoefficientlyimplementanarbitraryunitaryinterms
of two-qubit gates [6]; our numerical results suggest a
FIG. 5. Optimal number of entangling gates needed
similarresultforN-qubitgates. Inthetwo-qubitcasethe
to compile random Clifford operations, for sample sizes
compilation always succeeded with 3 entangling gates, {1000,200,100} for N = {2,3,4} qubits. Each Clifford gate
and not less (using 200 search repetitions). This was to wasgeneratedusing10N8stepsoftherandomwalkdescribed
be expected, since for two qubits an MS gate is equiva- inRef.[31]. Errorbarscorrespondtoonestandarddeviation.
lenttoaCNOTgate,anditisknownthat3CNOTgates
are enough (and in general necessary) to implement an
arbitrary two-qubit unitary [27, 28]. In the three-qubit
case, the optimization always succeeded with 8 entan-
gling gates, and never with fewer (also using 200 repeti-
tions). For 4 qubits, the optimization always succeeded
for 25 entangling gates, and succeeded only 4% of the
C. Compilation of isometries
time with 24 entangling gates. However, we did only
4 optimization repeats in the four-qubit case, owing to
the increased time it takes for these to converge. There- Aparticularcaseofinterestisthecompilationofauni-
fore,itmightbethecasethatgivenenoughoptimization tary whose action only matters on certain input states.
runs,moreunitarieswouldhavebeencompiledwithonly Thishappens,forinstance,whenoneisinterestedinstate
24 gates. We are not aware of any result in the litera- preparation starting from some fixed input state. Such
tureconcerningthenumberofN-qubitglobalentangling operationsbelongtothemoregeneralclassofoperations
gates required for implementing a general N-qubit uni- known as isometries [14, 30]. In this case, the problem
tary for more than N = 2 qubits. From our numerical to be solved has less constraints than when fully speci-
results, we conjecture that any three-qubit unitary can fying the target unitary, so a simpler sequence may be
beimplementedusingatmost8MSgates, andanyfour- found. In this section we focus on compiling a unitary
qubit unitary using at most 24 or 25 MS gates. thatisonlyspecifiedinaparticularsubspaceoftheinput
7
states, for example: where the sum goes over all the subspaces with different
relativephases,andU| isarectangularmatrixwiththe
 . .  Sj
u u .. .. components of the unitary in the j-th subspace.
11 12
 
u21 u22 free free
Utarget = .. .. , (26)
u31 u32 . .  D. Compensation of systematic errors
 . . 
. .
u u . .
41 42
Owingtosystematicerrors,theoperationsexperimen-
where the columns marked as ‘free’ are left unspecified. tally applied may still be unitary but deviate from the
Inthiscase, asuitablefidelityfunctionforthenumerical intendedones. Anexampleofthisisaddressingcrosstalk
optimization is: duetolaserlightleakingontoadjacentqubits. Ifitispos-
(cid:12) (cid:16) (cid:17)(cid:12)2 sible to characterize the actual experimental operations
f(U)=(cid:12)(cid:12)tr U|SUtarget|†S (cid:12)(cid:12) , (27) being applied, then they can be taken into account for
thecompilationbyadaptingouroptimizationprocedure:
where U| is a rectangular matrix with the components
S
of the unitary in the restricted subspace. 1. Compile the target unitary in terms of the ideal
A more general case is where some of the relative gates.
phases of the projections of the unitary acting on differ-
2. Replacetheidealgatesbytheexperimentallychar-
ent subspaces of the whole Hilbert space are irrelevant.
acterized operations.
For example, suppose that one wants to apply a unitary
to map some observable onto an ancilla qubit and then
3. Add operations to obtain a higher fidelity with the
measuretheancilla,asshowninfigure6. Sincetheinput
ideal target unitary.
state of qubit 3 is known to be |0(cid:105), only the subspace of
inputstatesspannedby{|000(cid:105),|010(cid:105),|100(cid:105),|110(cid:105)}isrel- Asanexampleweshowthatexcessivecrosstalkcanbe
evant. Moreover, the measurement will project the state corrected in an implementation of a Toffoli gate. Figure
ofthesystemontoeitherthesubspacespannedby{|000(cid:105), 7 depicts experimental data corresponding to the action
|010(cid:105), |100(cid:105)}, or that spanned by {|111(cid:105)}, and all phase of the Toffoli gate on the 8 input basis states. It can be
coherencebetweenthesealternativeswillbelost. There- seen that, by adding just two pulses, the output fidelity
fore, the compiled sequence can be sought such that it for each input state increased in some cases by up to
matches the desired unitary in each of the subspaces but 20%. Thesequencewith11pulsesisactuallyonlyanap-
allowing an arbitrary phase φ between them: proximate correction to the uncorrected case. The exact
 . . . .  correction requires 14 pulses, and actually yields a lower
. . . .
1 . 0 . 0 . 0 . fidelity than the approximate one, since it requires more
 .. .. .. ..  pulsesandeachofthesehasanon-zeroerrorprobability.
0 . 0 . 0 . 0 . 
 . . . . 
 . . . . 
0 . 1 . 0 . 0 . 
0 free 0 free 0 free 0 free IV. CONCLUSIONS AND OUTLOOK
Utarget = .. .. .. ..  (28)
0 . 0 . 1 . 0 . 
0 ... 0 ... 0 ... 0 ...  tumInuthniistawrioerskinwteohaavseeqsuheonwcne omfectohloledcstitvoecroomtaptiiloenqs,uaadn--
 .. .. .. ..  dressed rotations and global entangling operations. For
0 . 0 . 0 . 0 .  local unitaries, we have demonstrated an analytic ap-
 
. . . .
0 .. 0 .. 0 .. eiφ .. proach that produces the shortest possible sequences in
thegeneralcase,andadaptedthemethodtosimplifythe
In this case (figure 6) it is possible to find a simpler im- resultingsequencesifsomeconstraintsontheunitaryare
plementation than in the fully constrained case (figure lifted. For arbitrary unitaries, we have presented an ap-
2), owing to the additional degrees of freedom available, proach that produces sequences of layered local and en-
namely arbitrary outputs for the |ψ (cid:105) = |1(cid:105) input states tanglingoperations. This approachisbasedonanumer-
3
and an arbitrary relative phase between the two possible icaloptimizationprocedurethatisfasterthanpreviously
measurement outcomes. usedones,andthesequencesobtainedarebydesignopti-
In the general case considered here we want to maxi- malwithrespecttothenumberofentanglinggates. Our
mize the fidelity in each subspace, without regard to the numerical results suggest upper bounds on the number
relative phases between these. Therefore we can seek to of N-qubit gates required to implement arbitrary three-
maximize the function f consisting of the sum of the fi- and four-qubit unitaries.
delity functions (27) corresponding to each subspace: The results of this paper show that in many cases one
may obtain more efficient implementations by consider-
(cid:88)(cid:12) (cid:16) (cid:17)(cid:12)2
f(U)= (cid:12)tr U| U |† (cid:12) , (29) ing operations more general than two-qubit entangling
(cid:12) Sj target Sj (cid:12) gates. However, theexponentiallygrowingcomplexityof
j
8
FIG. 6. Left: a unitary mapping (Toffoli gate) is applied, after which qubit 3 is measured. Right: a pulse sequence for
implementing the circuit on the left. This implementation is simpler than in the fully constrained case (figure 2) because of
the additional degrees of freedom when compiling.
FIG. 7. State fidelity for a Toffoli gate applied on the 8 canonical input states.
decompositions as the number of qubits increases points Appendix A: Compiling local unitaries
to the necessity of keeping the register size small.
1. Finding basis changes
In this appendix we will show how to satisfy equation
V. ACKNOWLEDGEMENTS
(14). We need to find a rotation C around the equator
of the Bloch sphere such that:
We thank I. Chuang, B. Lanyon, and V. Nebendahl
for fruitful discussions. We thank the referees for bring- u=C−1σ C, (A1)
z
ing Refs. [14, 30–32] to our attention. We gratefully ac-
knowledgesupportbytheAustrianScienceFund(FWF), whereuisthegeneratorofagivenknownunitaryU,and
throughtheSFBFoQuS(FWFProjectNo. F4002-N16), it can always be written as:
as well as the Institut fu¨r Quantenoptik und Quantenin-
formationGmbH.E.A.M.isarecipientofaDOCfellow- u=sinθcosφσ +sinθsinφσ +cosθσ , (A2)
x y z
ship from the Austrian Academy of Sciences. P.S. was
supported by the Austrian Science Foundation (FWF) for some angles θ, φ.
Erwin Schr¨odinger Stipendium 3600-N27. This research In general C is of the form:
was funded by the Office of the Director of National
Intelligence (ODNI), Intelligence Advanced Reasearch C =e−iγc/2, (A3)
Projects Activity (IARPA), through the Army Research
Office grant W911NF-10-1-0284. All statements of fact, where γ is its rotation angle and c its generator, which
opinion or conclusions contained herein are those of the mustlieontheequatorandthusbealinearcombination
authors and should not be construed as representing the of σ and σ . If we propose:
x y
officialviewsorpoliciesofIARPA,theODNI,ortheU.S.
Government. c=sinφσ −cosφσ , (A4)
x y
9
and replace in equation (A1), we find that the angle of provided by Z(cid:48). We can now follow the same steps as in
rotation must be: section II. The unitary C is given by:
N
γ =θ. (A5) CN =Z(cid:48)−1UNC1−1C2−1···CN−1−1, (A15)
andeliminatingthisfactorfromtherestoftheequations
we obtain:
2. Writing a unitary as a product of two equatorial
rotations
U−1U =C−1Z C , (A16)
N 1 1 1 1
U−1U =C−1C−1Z C C ,
Wewillshowherehowtodecomposeanarbitraryuni- N 2 1 2 2 2 1
tary as a product of two rotations around the equator of .
.
.
the Bloch sphere, namely:
U−1U =C−1C−1···C−1 Z C ···C C .
N N−1 1 2 N−1 N−1 N−1 2 1
U =C C . (A6)
2 1
Equations (A16) can be satisfied in exactly the same
The target unitary can be written as: way as explained in section II. In order to satisfy equa-
tion (A15) we need to find a rotation Z(cid:48) such that the
(cid:18) (cid:19) (cid:18) (cid:19)
β β generatorofC liesontheequator. Thiscanbedoneas
U =cos 1−isin × N
2 2 follows.
We wish to find how to satisfy equation (A15). For
×(sinθcosφ σ +sinθsinφ σ +cosθ σ ), (A7)
x y z
this we need to find a rotation Z around the Z axis and
where β is its rotation angle,and θ,φ determine its ro- a rotation C around an axis on the equator of the Bloch
tation axis. Similarly, the equatorial rotations can be sphere such that, for a given unitary U, the following
written as: equation holds:
C =cos(cid:16)αi(cid:17)1−isin(cid:16)αi(cid:17)(cosφ(cid:48) σ +sinφ(cid:48) σ ), (A8) C =ZU. (A17)
i 2 2 i x i y
U is in general of the form:
for some rotation angles α and phases φ(cid:48).
i i
We shall asume that: U =e−iαu/2, (A18)
α =α =α, (A9) and Z is of the form:
1 2
φ(cid:48)1 =φ+∆/2, (A10) Z =e−iβσz/2. (A19)
φ(cid:48) =φ−∆/2. (A11)
2
We will first find the angle of rotation β. If we write
Replacing these into (A6) and solving for α and ∆ we out(A17)intermsofthegeneratorsofU andZ wehave:
obtain:
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19) (cid:19)
β β
(cid:16)α(cid:17) 1(cid:18) (cid:18)β(cid:19) (cid:19) C = cos 2 1−isin 2 σz ×
cos2 = cos +1 sin2θ, (A12)
2 2 2 (cid:16) (cid:16)α(cid:17) (cid:16)α(cid:17) (cid:17)
× cos 1−isin u . (A20)
cos2(cid:0)α(cid:1)−cos(cid:16)β(cid:17) 2 2
2 2
cos∆= 1−cos2(cid:0)α(cid:1) . (A13) Since the axis of rotation of C lies on the equator, its
2 generator must not have any Z component, and thus:
(cid:18)β(cid:19) (cid:16)α(cid:17) (cid:18)β(cid:19) (cid:16)α(cid:17)
0=sin cos +cos sin u , (A21)
3. Unitaries up to a collective Z rotation 2 2 2 2 z
Suppose that the unitary U we want to implement that is:
is followed by gates whose phase can be freely chosen. (cid:16) (cid:16)α(cid:17) (cid:17)
β =−2arctan tan u . (A22)
Then it must only be specified up to an arbitrary col- 2 z
lective rotation Z(cid:48), since this phase can be absorbed in
Once β is known, the unitary on the right-hand side of
the following gates. To compile U, we shall consider a
(A17) is fully determined, and thus C as well.
decomposition of the form (7):
U =Z(cid:48)C Z C ···Z C Z C . (A14)
N N−1 N−1 2 2 1 1
4. Unitaries up to independent Z rotations
Such adecomposition ismore convenient inthis case be-
cause the last addressed pulse Z has been eliminated Finally, suppose that the unitary we want to imple-
N
by taking advantage of the additional degree of freedom mentisdefineduptoarbitraryindependentrotationsfor
10
each qubit around the Z axis. This is useful if the uni- In order to solve equation (A28) we need to find ro-
tary is followed by a projective measurement, since any tations Z = Z(β ), Z = Z(β ) that satisfy a general
1 1 2 2
finalrotationaroundthemeasurementaxisforanyqubit equation of the form:
simply adds a phase and will not change the measured
probabilities. [Z1U1,Z2U2]=0, (A29)
Let us again consider a sequence of the form (7). The
for given arbitrary U , U , whose generators are u and
decomposition must now satisfy, for each qubit: 1 2 1
u respectively.
2
Z(cid:48)U =C ···C Z C , (A23) Let us define:
1 1 N 2 1 1
Z2(cid:48)U2 =CN···Z2C2C1, Vi =ZiUi, (A30)
.
.. and let vi be the generators of the Vi. In order to satisfy
Z(cid:48) U =Z C ···C C , (A29), the vi must satisfy:
N N N N 2 1
v =v =v, (A31)
where the Z(cid:48) are arbitrary rotations around the Z axis. 1 2
i
As before, we can set ZN =1 and find CN: since if two unitaries commute their generators must be
the same. Our first goal is to determine the generator v.
C =Z(cid:48) U C−1C−1···C−1 . (A24)
N N N 1 2 N−1 Let us consider the unitary:
EliminatingCN fromtheremainingequationsweobtain: W =Z1/2U Z1/2. (A32)
i i i i
UN−1ZN(cid:48)−1Z1(cid:48)U1 =C1−1Z1C1, (A25) By writing down Wi explicitly in terms of the genera-
UN−1ZN(cid:48)−1Z2(cid:48)U2 =C1−1C2−1Z2C2C1, tors of each factor, it can be seen that its generator wi
satisfies:
.
.
.
{w ,[σ ,u ]}=0. (A33)
i z i
U−1Z(cid:48)−1Z(cid:48) U =C−1···C−1 Z C ···C .
N N N−1 N−1 1 N−1 N−1 N−1 1
Since we have:
Eachequationhasnowanextradegreeoffreedomcom-
ingfromtheangleoftheZk(cid:48) rotation. Letusforsimplic- Vi =Zi1/2WiZi−1/2, (A34)
ity consider the case where the number of qubits N is
from equation (A33) we see that:
odd. Ifwegroupequations(A25)inpairswegettwode-
grees of freedom per pair, which can be used to remove (cid:110) (cid:111)
v,Z1/2[σ ,u ]Z−1/2 =0. (A35)
one of the global operations. Therefore we will discard i z i i
every even-numbered global operation C from our de-
2k
Thegeometricalmeaningofthisequationisthatthevec-
composition and look for the solution of the following
tor defined by v on the Bloch sphere is perpendicular to
system of equations:
thatdefinedbyZ1/2[σ ,u ]Z−1/2. Since(A35)musthold
i z i i
U−1Z(cid:48)(cid:48)U =C−1Z C , (A26) for i = 1,2, v must correspond to the cross product of
N 1 1 1 1 1
these vectors:
U−1Z(cid:48)(cid:48)U =C−1Z C ,
N 2 2 1 2 1 (cid:104) (cid:105)
UN−1Z3(cid:48)(cid:48)U3 =C1−1C3−1Z3C3C1, v =N Z11/2[z,u1]Z1−1/2,Z21/2[σz,u2]Z2−1/2 , (A36)
U−1Z(cid:48)(cid:48)U =C−1C−1Z C C ,
N 4 4 1 3 4 3 1 where N is chosen such that:
.
.
. 1
tr(v2)=1. (A37)
U−1Z(cid:48)(cid:48) U =C−1···C−1 Z C ···C , 2
N N−2 N−2 1 N−2 N−2 N−2 1
U−1Z(cid:48)(cid:48) U =C−1···C−1 Z C ···C , Having found v, it remains to find the rotation angles
N N−1 N−1 1 N−2 N−1 N−2 1
β . Now, v must satisfy [Z U ,v]=0, and therefore:
i i i
where Zk(cid:48)(cid:48) =ZN(cid:48)−1Zk(cid:48). If the number of qubits N is even, U vU−1 =Z(β )−1vZ(β ). (A38)
then the last equation is simply left unpaired. It is easy i i i i
to verify that for each pair of equations the right-hand BothvandU areknown,sovandU vU−1canbewritten
i i i
sides commute, and therefore we must have:
down explicitely as:
[UN−1Z2(cid:48)(cid:48)k−1U2k−1,UN−1Z2(cid:48)(cid:48)kU2k]=0, (A27) v =sinθcosφσx+sinθsinφσy+cosθσz,
(A39)
or equivalently:
U vU−1 =sinθcosφ(cid:48)σ +sinθsinφ(cid:48)σ +cosθσ ,
i i i x i y z
[Z(cid:48)(cid:48) U U−1,Z(cid:48)(cid:48) U U−1]=0. (A28) (A40)
2k−1 2k−1 N 2k 2k N