High-fidelity CZ gate for resonator-based superconducting quantum computers

0.59 MB·English

by  Joydip Ghosh

High-fidelity CZ gate for resonator-based superconducting quantum computers
Joydip Ghosh,1,∗ Andrei Galiautdinov,1,2 Zhongyuan Zhou,1
Alexander N. Korotkov,2 John M. Martinis,3 and Michael R. Geller1,†
1Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA
2Department of Electrical Engineering, University of California, Riverside, California 92521, USA
3Department of Physics, University of California, Santa Barbara, California 93106, USA
(Dated: January 10, 2013)
A possible building block for a scalable quantum computer has recently been demonstrated [M.
Mariantoni et al., Science 334, 61 (2011)]. This architecture consists of superconducting qubits
capacitively coupled both to individual memory resonators as well as a common bus. In this work
westudyanaturalprimitiveentanglinggateforthisandrelatedresonator-basedarchitectures,which
3 consistsofacontrolled-σz (CZ)operationbetweenaqubitandthebus. TheCZgateisimplemented
1 with the aid of the non-computational qubit |2i state [F. W. Strauch, et al., Phys. Rev. Lett. 91,
0 167005 (2003)]. Assuming phase or transmon qubits with 300MHz anharmonicity, we show that
2 by using only low frequency qubit-bias control it is possible to implement the qubit-bus CZ gate
with99.9% (99.99%) fidelityinabout17ns(23ns)witharealistictwo-parameterpulseprofile,plus
n
twoauxiliary z rotations. Thefidelitymeasurewerefertohereisastate-averaged intrinsicprocess
a
fidelity,which does not include any effects of noise or decoherence. These results apply to a multi-
J
qubit device that includes strongly coupled memory resonators. We investigate the performance
8 of the qubit-bus CZ gate as a function of qubit anharmonicity, indentify the dominant intrinsic

error mechanism and derive an associated fidelity estimator, quantify the pulse shape sensitivity
]
h and precision requirements, simulate qubit-qubitCZ gates that are mediated by the busresonator,
p and also attempt a global optimization of system parameters including resonator frequencies and
- couplings. Our results are relevant for a wide range of superconducting hardware designs that
nt incorporateresonators and suggest that itshould bepossibletodemonstratea99.9% CZ gatewith
a existingtransmonqubits,whichwouldconstituteanimportantsteptowardsthedevelopmentofan
u error-corrected superconducting quantumcomputer.
q
[ PACSnumbers: 03.67.Lx,85.25.Cp

1
v
I. QVN ARCHITECTURE both to individual memory resonators as well as a com-
9
mon bus, as illustrated in Fig. 1. The crossed boxes
1
7 Reachingthefidelitythresholdforfault-tolerantquan- in Fig. 1 represent the phase qubits [35] employed by
1 tumcomputationwithsuperconductingelectricalcircuits Mariantoni et al. [33], however other qubit designs such
1. [1–3] will probably require improvement in three areas: as the transmon may be used here as well. The key fea-
0 qubit coherence, readout, and qubit-qubit coupling tun- ture of this architecture is that information (data) is
3 ability. Fortunately, the coherence times of supercon- stored in memory resonators that are isolated by two
1 ducting transmon qubits [4, 5] have increased dramat- detuned coupling steps from the bus. Qubits are used
v: ically, exceeding 10µs in the three-dimensional version to transfer information to and from the bus or entangle
i [6, 7]. Fast, threshold-fidelity nondestructive measure- with it, and to implement single-qubit operations, but
X menthasnotyetbeendemonstrated,butisbeingactively are otherwise kept in their ground states. No more than
r pursued[8–12]. Somemethodforturningoffthe interac- one qubit (attached to the same bus) is to be occupied
a
tion between device elements—beyond simple frequency at any time. Such an approach significantly improves
detuning—is also desirable for high-fidelity operations. the effective on/off ratio without introducing the added
A variety of tunable coupling circuits have been demon- complexity of nonlinear tunable coupling circuitry. The
strated[13–19],but theseconsiderablyincreasethe com- spectral crowding problem of the usual qubit-bus archi-
plexity of the hardware and it is not clear whether they tecture is greatlyreducedbecause the four-stepcoupling
willbepracticalforlargescaleimplementation. Thecou- between memory resonators is negligible. And an added
pling can also be controlled by the application of mi- benefitoftheQVNapproachisthatthelongercoherence
crowave pulses [20–32]. timesofthememoryelementsreducetheoveralldecoher-
An alternative approach has been introduced by ence rate ofthe device. (In Ref. [34], the architecturewe
Mariantoni et al. [33] and theoretically analyzed in consider is referred to as the resonator-zero-qubit archi-
Ref. [34]. In this quantum von Neumann (QVN) archi- tecture, but here we will follow the QVN terminology of
tecture, superconducting qubits are capacitively coupled [33].)
The QVN architecture of Mariantoni et al. [33] is not,
by itself, capable of large-scale, fault-tolerant quantum
∗ [email protected] computation, nor is it knownhow multiple QVN devices
† [email protected] might be integrated into a scalable design. The problem
2
7.5
7.4
7.3
Hz)7.2
G
y (7.1
c
n
e7.0
u
q
e6.9
bit fr6.8
u
q
6.7
FIG.1. Layoutofthefour-qubitQVNprocessor. Theqi rep- 6.6
resent superconducting qubits capacitively coupled to mem-
ory resonators mi as well as a resonator bus b. 6.50 2 4 6 8 10 12 14 16
t (ns)
of designing scalable, fault-tolerant architectures for su- FIG. 2. Two-parameter CZ pulse profile (1) for the case
perconducting qubits is of greatinterest and importance of ω /2π = 6.8GHz, ω /2π = 7.5GHz, t = 7ns,
on off ramp
[36, 37], but is still in its infancy. We expect the gate σ = 1.24ns, and t = 10ns. The total gate time excluding
on
design approach discussed here to be applicable to fu- auxiliary z rotations is tgate = 17ns. The example shown
turearchitecturedesignsincorporatingqubitscoupledto is representative of a 99.9% fidelity gate for a qubit with
300MHzanharmonicity.
resonators,and perhaps beyond.
Along with high-fidelity single-qubit rotations [38,
39], quantum computing with the QVN processor re-
quires two additional types of operations. The first ory resonators, and the bus), zero-amplitude conditions
is state transfer between the different physical compo- must be enforced on the additional n 1 qubits, lead-
−
nents, which has to be performed frequently during a ing to a total of 2(n+1) pulse parameters, plus one z
computation. The simplest case of state transfer is be- rotation angle. This makes quasi-exact state transfer to
tweenaqubitanditsassociatedmemory(orthereverse). and from the bus a considerably more challenging op-
This case is investigated in Ref. [34], where two im- eration. Simpler three-parameter approximate transfers,
portant observations are made: First, in contrast with however,canstillbeimplementedwithveryhighfidelity,
the usual SWAP or iSWAP operation, which must be even when the coupling is strong (see below).
able to simultaneously transfer quantum information in Quantum computing with the QVN system also re-
two directions, only unidirectional state transfers are re- quires a universalSU(4) entangling gate, the most natu-
quired in the QVN system. This is because adjacent ralbeing acontrolled-σz (CZ)operationbetweenaqubit
qubits and resonators are—by agreement—never simul- and the bus. The CZ gate investigated here makes es-
taneously populated. Second, the phase of a transferred sential use of the non-computational qubit 2 state and
| i
1 state is immaterial, as it can always be adjusted by was first proposed by Strauch and coworkers [42]. The
| i
future qubit z rotations [40, 41]. These two simplifica- Strauchgatehasbeeninvestigatedby manyauthorsand
tionsallowtheresultingstatetransferoperation,calleda has been demonstrated in severalsystems [43–48].
MOVEgate,tobecarriedoutwithextremelyhighintrin- The present paper extends previous workby consider-
sicfidelity—perfectlyforatruncatedmodel—withasim- ing device parameters and pulse shapes appropriate for
ple four-parameter pulse profile. By intrinsic fidelity we the QVN system, and by optimizing the CZ fidelity in
mean the process fidelity (defined below) in the absence a multi-qubit device. We are especially interested in
of noise or decoherence. The need for four control pa- whether the absence of an active tunable coupler re-
rameters immediately follows from the requirement that sults in any significant limitation on the obtainable fi-
afteraMOVEgate,theprobabilityamplitudesmustvan- delity,givenareasonableamountofqubitcoherence,and
ish on two device components, the component (q or m) whether veryhighintrinsic fidelities canbe reachedwith
thestateisleaving,andthebusb. Eachzeroimposestwo a simple and experimentally realistic (filtered rectangu-
realparameters,andnootherprobabilityamplitudesac- lar)pulseshape. Wealsostudyhowthegateperformance
quire weight (in the truncated model). Fixing the phase rapidly improves with increasing qubit anharmonicity,
oftheMOVEgate,ifdesired,requiresoneadditionalcon- show that the dominant intrinsic error is caused by a
trol parameter in the form of a local z rotational angle. nonadiabatic excitation of the bus 2 state that occurs
| i
State transfer between a qubit and the bus (or the re- duringtheswitchingthequbitfrequency,deriveafidelity
verse)canbe analyzedin the same way,althoughin this estimatorbasedonthatmechanism,analyzepulseshape
case more pulse-shape parameters are required. In an errors, and simulate qubit-qubit CZ gates mediated by
n-qubit QVN processor (consisting of n qubits, n mem- the bus. Finally, we address the interesting problem of
3
system optimization, by using gate and idling error esti- pulse-shape parameters and two are auxiliary local z ro-
mates to deduce optimal values of resonator frequencies tation angles. We emphasize that only low-frequency
and couplings. pulses are required, and that the number of control pa-
rameters does not depend on the number of qubits in
the QVN device. The results quoted above assume four
phase or transmon qubits with 300MHz anharmonicity;
II. SUMMARY OF RESULTS
othervaluesofanharmonicityareconsideredbelow. The
CZ gate referredto here is between qubit q and the bus
1
We find that veryhigh intrinsic fidelites—in the range (see Fig. 1), not between a pair of qubits as is usually
of 99.9% to 99.99% and with corresponding total gate considered.
timesintherangeof17to23ns—canindeedbeobtained The two-parameter low frequency pulse profile we use
with a four-parametergate. Two controlparameters are throughout this work is
ω ω t 1t t t + 1t
ǫ(t)=ω + on− off Erf − 2 ramp Erf − gate 2 ramp , (1)
off
2 √2σ − √2σ
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
an example of which is shown in Fig. 2. Here ǫ is the qubit frequency, ω and ω are the frequencies off and near
off on
resonance (with the bus), and the pulse switching time is determined by σ, the standard deviation of the Gaussians
inside (1). The value of t determines how the pulse is truncated at t = 0 and t as explained in Sec. IIIF;
ramp gate
throughout this work we assume that
t =4√2σ. (2)
ramp
The relation (2) allows the switching time to be alternatively characterizedby t , which, as Fig. 2 illustrates, is a
ramp
measure of the width of the ramps. The variable t is the total execution time of the gate excluding z rotations.
gate
The two control parameters ω and
on
t t t (3)
on gate ramp
≡ −
aredeterminedbythe numericaloptimizationproceduredescribedinSec.IIIG.From(3)weinferthatt isthe time
on
intervalbetweenthe midpoints ofthe ramps,orthe full-width at half-maximum(FWHM) ofthe pulse. We note that
the optimal values of t are somewhat longer than the value
on
π
tsudden (4)
on ≡ √2g
b
that applies in the sudden, σ 0 limit. Inaddition to ω and t , two auxiliarylocal z rotations—onthe qubit and
on on
resonator—are used to implem→ent the CZ gate. As we explain below, adjusting the two control parameters ω and
on
t zeros the population left in the non-computational qubit 2 state after the gate and (along with the auxiliary z
on
| i
rotations)sets the controlledphase equal to 1. The pulse shape (1) describes a rectangularcurrentor voltagepulse
−
sentto the qubit frequency bias througha Gaussianfilter ofwidth σ, andis believed to be anaccurate(althoughnot
exact) representation of the actual profile seen by the qubits in Ref. [33].
TABLE I. Optimal state-averaged process fidelity F for the Strauch CZ gate between qubit q and the bus, in the QVN
ave 1 4
processor of Fig. 1. No decoherence or noise is included here. Specifications for 99.9% and 99.99% gates are provided for
threevaluesofqubitanharmonicityη. Theparameterst andσ characterizethepulseswitchingtime,andt isthetotal
ramp gate
gate time excludingauxiliary z rotations. F is the minimum fidelity. Data after double vertical lines give thenonadiabatic
|11i
switching error and minimum fidelity estimates; these quantitiesare defined and discussed in Sec. IIIH.
η/2π g /2π g /2π tsudden t σ t t F F |A|2 p F(est)
b m on ramp on gate ave |11i sw |11i
200MHz 30MHz 100MHz 11.8ns 11ns 1.94ns 15.8ns 26.8ns 99.901% 99.613% 2.1×10−2 1.5×10−3 99.692%
16ns 2.83ns 18.3ns 34.3ns 99.992% 99.975% 2.8×10−3 2.0×10−4 99.960%
300MHz 45MHz 100MHz 7.9ns 7ns 1.24ns 9.9ns 16.9ns 99.928% 99.714% 1.7×10−2 1.2×10−3 99.761%
11ns 1.94ns 11.8ns 22.8ns 99.995% 99.979% 9.9×10−4 7.2×10−5 99.986%
400MHz 60MHz 100MHz 5.9ns 5ns 0.88ns 7.0ns 12.0ns 99.950% 99.804% 1.4×10−2 1.0×10−3 99.799%
7ns 1.24ns 7.8ns 14.8ns 99.991% 99.966% 2.1×10−3 1.5×10−4 99.970%
Our main results are given in Table I. Here η is the qubit anharmonicity. The 200MHz results apply to the
4
phase qubits of Ref. [33], while the larger anharmonic- line with that required for fault-tolerant quantum com-
ities might be relevant for future implementations with putationwith topologicalstabilizer codes [49–51]. Qubit
transmons. The bus couplings g are determined by the anharmonicityis animportantresourcethat will helpus
b
“g optimization” procedure described in Sec. V, which achieve that goal.
leads to the simple formula TheCZgateofTableIisbetweenqubitq andthebus
1
inthe QVN device,andbegins(typically)withasuper-
4
gb position of qubit-bus eigenstates, with the other qubits
=0.15, (5)
η and all memory resonators in their ground states. In
Sec. IV we also comment on severalextensions and vari-
for the (approximately) optimal bus coupling. Let ationsofthisbasicqubit-busCZgate: Tobeginwith,the
“QVN ”refertoaquantumvonNeumannprocessorwith samegate butwith qubitq (whichhasa differentmem-
n 4
n qubits coupled to n memory resonatorsand a bus; the oryfrequency)isconsideredinSec.IVA.InSec.IVBwe
Hamiltonian for such a device is discussed in Sec. IIIA. simulateaCZgatebetweentwoqubitsinQVN ,starting
4
As indicated in Table I, the memory resonators are al- in the idling configuration where the qubits are empty
ways strongly coupled to allow for fast (less than 5ns) and all data is stored in memory. In this case the qubit-
MOVE operations in and out of memory. CZ fidelities busCZgateofTableIissupplementedwithMOVEgates
well above 99.99% are also obtainable (see Sec. IIIJ). to effect a CZ between qubits. And in Sec. IVC we dis-
Table I shows that the time required for a qubit-bus CZ cuss the CZ implemented between a pair of directly cou-
gate with fixed intrinsic fidelity is inversely proportional pledanharmonicqubits,insteadofaqubitandresonator.
to the qubit anharmonicity, namely This is the system originally considered by Strauch et
al. [42].
5.2 6.7
t(99.9%) and t(99.99%) . (6)
gate ≈ η/2π gate ≈ η/2π
III. CZ GATE DESIGN
These expressions indicate that CZ gates with very high
intrinsic fidelity can be implemented in about 20ns with Inthissectionwediscussthequbit-busCZgatedesign
existing superconducting qubits, a conclusion which ap- problem.
plies not only to QVN but also to a wide range of
n
similar resonator-based architectures. The intrinsic gate
(or process) fidelity F is the squared overlap of ideal A. QVN model
ave
and realized final states, averagedover initial states (see
Sec.IIIE).Byintrinsicwemeanthatnoiseanddecoher- The QVN processor consists of n superconducting
n
encearenotincludedinthegatesimulation. Thefidelity qubits[1–3]capacitivelycoupled[42,52,53]tonmemory
estimate is developed in Sec. IIIH. The results given in resonators and to a common bus resonator. Here we as-
Table I apply specifically to the n = 4 processor, but sumeparametersappropriateeither forphasequbits [35]
similar results are expected for other (not too large)val- ortransmonqubits[4,5]withtunabletransitionfrequen-
ues of n. Two strategies are critical for obtaining this cies. We write the qubit angular frequencies as ǫ , with
i
high performance: Separating two control parameters i=1, ,n. These are the only controllable parameters
···
in the form of auxiliary local z rotations, and defining in the QVN Hamiltonian (in contrast with Refs. [20–32]
the computational states in terms of interacting system we do not make use of microwave pulses). The memory
eigenfunctions. These strategies were used in Ref. [34] frequencies are written as ω , and the bus frequency is
mi
and are discussed in more detail below. The gate fideli- ω . The (bare) frequencies of all resonatorsare assumed
b
tiesachievablewithatransmon-basedQVNdevicearein here to be fixed.
Because we are interested in very high fidelities, a realistic model is required. However, we have shown (in un-
published work) that the CZ performance is extremely robust with respect to the model details, so we only report
results for a simplified Hamiltonian; the approximationsused are discussed below. For the qubit-bus CZ simulations,
the Hilbert space is truncated to allow for up to three excitations. The CZ gate naively involves no more than two
excitations, so to properly account for leakage we include up to three. Therefore, four-level qubits and resonators
(which include the 3 states) are required in the model. The QVN Hamiltonian we use is
| i
0 0 0 0 0 0 0 0 0 0 0 0
n
0 ǫ 0 0 0 ω 0 0 0 ω 0 0
H = 0 0i 2ǫi η 0  +0 0mi 2ωmi 0  +gmYqi⊗Ymi+gbYqi⊗Yb+0 0b 2ωb 0 ,
Xi=10 0 0− 3ǫi−η′qi 0 0 0 3ωmimi  0 0 0 3ωbb
      (7)
5
excluding single-qubit terms for microwave pulses that B. Strauch CZ gate
are not used in this work. Here η and η′ are qubit an-
harmonic detuning frequencies, gm andgb are the qubit- Inthis sectionwe giveadetaileddescriptionofthe CZ
memory and qubit-bus interation strengths, and gate introduced by Strauch et al. [42]. In particular, we
explain the specific roles played by the two pulse-shape
parameterst andω ,andby the two auxiliaryz rota-
0 i 0 0 on on
i −0 √2i 0 tion angles γ1 and γ2. To accomplish this we introduce
Y  − . (8) several approximations that allow for an analytic treat-
≡ 0 √2i 0 −√3i ment of the CZ gate dynamics.
0 0 √3i 0 
  First, we consider a truncated model consisting of a
  single superconducting qubit with frequency ǫ and an-
harmonic detuning η, capacitively coupled to a bus res-
The matrices in (7) act nontrivially in the spaces indi-
onator with frequency ω ,
cated by their subscripts, and as the identity otherwise. b
The matrix Y resultsfroma harmonicoscillatorapprox- 0 0 0 0 0 0
imation for the qubit eigenfunctions. Factors of ~ are H = 0 ǫ 0 + 0 ω 0 +g Y Y . (9)
b b q b
suppressed throughout this paper. 0 0 2ǫ η 0 0 2ω  ⊗
b
− q b
Themainapproximationsleadingto(7)aretheneglect    
In this case Y reduces to
of the ǫ-dependence of the interaction strengths g and
m
gb,andtheneglectofasmalldirectcouplingbetweenthe 0 i 0
memories and bus [54]. We have verified that including Y = i −0 √2i . (10)
these does notchangethe mainconclusions ofthis work. 0 √2i −0 
The ǫ-dependence of the anharmonicities η and η′, and
 
small anharmonic correctionsto the interactionterms in This Hamiltonian is written in the basis of bare eigen-
(7), are also neglected. states, which are the system eigenfunctions when the
qubit and resonator are uncoupled. We write these bare
The parameter values we use in our simulations are
states as qb , with q,b 0,1,2 . The energies of the
providedinTableII.Weassumeη′ =3η,whichisappro- | i ∈ { }
interacting eigenstates, which we write with an overline
priate for qubic anharmonicity. As discussed in Sec. V,
as qb , are plotted in Fig. 3 as a function of ǫ/2π for
sthheortveasltueCoZf gtahteebtuims eco(ufoprlinagrgabngies cohfofisednelittoiesg)i.veTthhee the|caise of ωb/2π = 6.5GHz, η/2π = 300MHz, and
g /2π =45MHz. Theinteractingeigenstatesarelabeled
choiceofresonatorfrequenciesisalsodiscussedinSec.V. b
such that qb is perturbatively connected to qb when
We simulate n = 4 qubits. The fidelities quoted in
| i | i
ǫ ω .
this paper are numerically exact for the model (7); the b
≫
Second, we assume a short switching time and ignore
rotating-waveapproximation is not used.
the dynamical phases acquired during the ramps. As we
will see below, this approximation is valid when g η,
b
≪
so that the switching can be made sudden with respect
TABLE II. Deviceparameters used in thiswork. tothe couplingg ,butstilladiabaticwithrespectto the
b
quantity value anharmonicity η.
empty qubitparking frequency ωpark/2π 10.0GHz TheCZgateofStrauchetal.[42],adaptedtothequbit-
memory resonator m1 frequency ωm1/2π 8.3GHz resonator system, works by using the anticrossing of the
memory resonator m2 frequency ωm2/2π 8.2GHz 11 channel with the auxiliary state 20 . In terms of
memory resonator m frequency ω /2π 8.1GHz | i | i
3 m3 the pulse parameters defined in (1), the qubit-resonator
memory resonator m frequency ω /2π 8.0GHz
4 m4 state is prepared at a qubit frequency ǫ = ω , and the
initial detuned qubit frequency ω /2π 7.5GHz off
off frequency is then switched to ǫ=ω for a FWHM time
bus resonator frequency ω /2π 6.5GHz on
b duration t . In the simplified model considered in this
qubit-memory coupling strength g /2π 100MHz on
m
section,
qubit-buscoupling strength g /2π 30−60MHz
b
qubitanharmonicity η/2π 200−400MHz
ω =ω +η, (11)
on b
and
Although the CZ and MOVE gates considered here do π
t = . (12)
not involve microwave pulses, the single-qubit gates are on
√2g
b
assumedtobeimplementedwithmicrowavesintheusual
manner at the qubit frequency ω . This frequency is Equation (11) gives the qubit frequency for which the
off
also used to define an experimental “rotating” reference bare state 11 is degenerate with 20 , and is at a fre-
| i | i
frame or localclock for eachqubit: All qubit frequencies quency η above the usual resonance condition. Equa-
are defined relative to ω [33]. This is discussed below tion (12) is the sudden-limit value defined in (4) and de-
off
in Sec. IIIC. rived below. The qubit frequency is then returned to
6
The 01 component will mostly return to 01 , also
| i | i
with an acquired phase, but a small component will be
left in 10 due to the nonadiabatic excitation of that
| i
channel, which is only separated in energy from 01 by
| i
about η when ǫ=ω +η. The 10 component similarly
b
| i
suffers from a small noniadabatic coupling to 01 . As
| i
we will explain below, these nonadiabatic errors are ex-
14.8 20 ponentially suppressed when the functional form of ǫ(t)
14.4 is properly designed. Then we have
14.0 11
01 e−iα 1 E 01 +e−iα′ E 10 (16)
13.6 1 1
→ −
z) 13.2 02 and (cid:12)(cid:12) (cid:11) p (cid:12)(cid:12) (cid:11) p (cid:12)(cid:12) (cid:11)
H
12.8
G
( 12.4 10 e−iβ 1 E1 10 +e−iβ′ E1 01 , (17)
h → −
rgy/ 12.0 wher(cid:12)(cid:12)eE(cid:11)1 isasmpallnonadi(cid:12)(cid:12)aba(cid:11)ticpopuplation(cid:12)(cid:12)err(cid:11)or(below
ne 7.6 10 we referto E1 asa switching error). Inthe E1 0limit,
e 7.2 α and β are dynamical phases given by →
66..48 01 α= tgateE dt ω gb2 t , (18)
01 b on
≈ − η
6.0 Z0 (cid:18) (cid:19)
tgate g2
0 . 00 00 β = E10dt≈ ωb+η+ ηb ton, (19)
Z0 (cid:18) (cid:19)
6.50 6.75 7.00 7.25 7.50
where the second approximate quantities neglect phase
qubit frequency (GHz)
accumulationduring the ramps and use perturbative ex-
pressions for the energies E and E when ǫ=ω +η.
01 10 b
Theexpressions(16)and(17)neglectanextremelysmall
FIG. 3. (Color online) Energies of eigenstates |qbi of asingle
leakage out of the 01 , 10 subspace. Neglecting this
qubitq coupledtoa resonator busb. Hereω /2π =6.5GHz, {| i | i}
b leakage,the evolution in the 01 , 10 subspace is uni-
η/2π = 300MHz, and gb/2π = 45MHz. The time depen- {| i | i}
dence of the qubit frequency during a CZ gate (solid black tary, leading to the phase condition
curve) is indicated at thetop of thefigure.
ei(α−β′)+ei(α′−β) =0. (20)
the detuned value ω . The complete pulse profile is Using (20) to eliminate β′ leads to
off
also shown in Fig. 3 (solid black curve) for the case of
ωon/2π=6.8GHz and ωoff/2π=7.5GHz. 01 →e−iα 1−E1 01 +e−i(β+φ) E1 10 ,(21)
maLliezte’sd)foqlluobwitt-hreeseovnoaltuotriosntarteesultingfromaninitial(nor- (cid:12)(cid:12)10(cid:11)→e−iβp1−E1 (cid:12)(cid:12)10(cid:11)−e−i(α−φ)pE1 (cid:12)(cid:12)01(cid:11),(22)
wher(cid:12)(cid:12)e φ(cid:11) α′ pβ. The ev(cid:12)(cid:12)olu(cid:11)tion of thepeigent(cid:12)(cid:12)ate(cid:11)s 01
a00 00 +a01 01 +a10 10 +a11 11 . (13) ≡ − | i
and 10 isthereforecharacterizedbythecross-excitation
Because the(cid:12)(cid:12)00(cid:11) chann(cid:12)(cid:12)el(cid:11)is very(cid:12)(cid:12)wel(cid:11)l separ(cid:12)(cid:12)ate(cid:11)dfrom the prob|abiility E1 and three phase angles α, β, and φ.
| i Nowweconsiderthe 11 component. The 11 channel
others,the 00 componentwillonlyacquireadynamical | i | i
| i couples strongly with the 20 channel, as well as weakly
phase factor
| i
with 02 . The simplestwaytounderstandthe dynamics
e−iE00tgate, (14) | i
of the 11 component is to use two different represen-
| i
tations to describe these two effects. We will describe
where E is the energy of the 00 eigenstate. Without
00 strong interaction with 20 in the bare basis and the
lossofgeneralitywecanshifttheentirespectrumsothat | i
(cid:12) (cid:11) weak, nonadiabatic coupling with 02 in the eigenstate
E00 = 0 [as in (9)] and the ph(cid:12)ase factor (14) becomes basis. Supposewebeginwiththequ|biitstronglydetuned
unity. Thisfreedomresultsfromthefactthatanyunitary
fromthebus,sothat 11 11 (thedetunedinteracting
gate operation only needs to be defined up to an overall | i≈| i
eigenstate is well approximated by the bare 11 state).
multiplicative phase factor. With this phase convention | i
the CZ gate acts as the identity on this component, so Thenwequicklyswitchǫfromωoff toωb+η.By“quickly”
wemeanthatwestronglymixwiththe 20 channel. The
we have the map | i
interaction with 02 is always weak, even in the sudden
| i
00 00 . (15) limit. Thisasymmetricexcitationispossiblebecause 20
→ | i
(cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12)
7
isprotected(separatedinenergyfrom 11 )byanenergy are implemented in software (they are compiled into fu-
| i
gap 2√2g , whereas 02 is protected by a much larger ture qubit rotations). Following the pulse sequence that
b
gapofη √2g (this|expiressionaccountsforlevelrepul- leads to (15), (21), (22), and (31), with the operation
b
−
sion from 20 , and we have assumed that g η). We
| i b ≪ Rz(γ1) Rz(γ2), (33)
caninformallysaythat the desiredswitching is nonadia- ⊗
baticwithrespecttotheenergyscalegb,butisadiabatic where
with respect to η [42].
γ = β and γ = α, (34)
Focusing first on the strong coupling to 20 , the sud- 1 2
− −
| i
denlyswitched 11 stateisnolongeraneigenstatewhen
| i leads to the map
ǫ=ω ,astherelevanteigenfunctionsatthissettingare
on
00 00 , (35)
11 20 11 + 20 →
|11i= | i√−2| i and |20i= | i√2| i. (23) (cid:12)(cid:12)01(cid:11)→(cid:12)(cid:12) 1(cid:11)−E1 01 +e−iφ E1 10 , (36)
The nonstationary state (cid:12)(cid:12)10(cid:11)→p1−E1 (cid:12)(cid:12)10(cid:11)−eiφ pE1 0(cid:12)(cid:12)1 ,(cid:11) (37)
(cid:12)11(cid:11) p11 , (cid:12) (cid:11) p (cid:12) (cid:11) (38)
11 + 20 (cid:12) →− (cid:12) (cid:12)
11 = | i | i (24) apart fro(cid:12)m (cid:11)a globa(cid:12)l p(cid:11)hase factor. The use of auxiliary z
| i √2 (cid:12) (cid:12)
rotations is discussed further Sec. IIID.
The minus sign in (38) is the key to the Strauch CZ
therefore rotatesinthe 11 , 20 subspace, andafter a
time duration t becomes{| i | i} gate. However,as mentioned above,the analysis leading
to (38) neglected a weak nonadiabatic excitation of the
ψ =e−iE11t |11i+e−i∆Et|20i (25) |02i channel caused by the switching of ǫ. Including this
| i √2 effect in (38) leads to the modification
(cid:20) (cid:21)
1+e−i∆Et
=e−iE11t 11 11 1 E 11
2
2 →− −
(cid:20) (cid:18)1 e−i∆Et(cid:19)(cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) +pphase fact(cid:12)(cid:12)or(cid:11)× E2 02 , (39)
− 20 , (26)
−(cid:18) 2 (cid:19) (cid:21) whereE2isanotherswitchingerror.pBoth(cid:12)(cid:12)E1(cid:11)andE2van-
(cid:12) (cid:11) ish exponentially with σ (or t ), and for the regimes
where (cid:12) studied in this work E is therdaommpinant source of intrin-
2
sic gate fidelity loss. We note that the analysis leading
∆E E E =2√2g . (27)
≡ 20− 11 b to (39) assumed implementation of the ideal values [(11)
and (12)] of ω and t . Errors in these two control
Holding ǫ fixed at ω +η for a FWHM time (12), corre- on on
b
parameters, which we refer to as pulse shape errors and
sponding to a 2π rotation, (26) becomes
study in Sec. IIII, lead instead to
ψ =e−iE11ton 11 . (28)
| i | i 11 eiδ 1 E E 11
2 θ
→− − −
When ǫ=ωb+η, the energy of eigenstate |11i is (cid:12)(cid:12) (cid:11) +phapse factor× E(cid:12)(cid:12)2 (cid:11)02
E11 =2ωb+η √2gb. (29) +phase factor×pEθ (cid:12)(cid:12)20(cid:11), (40)
−
wherethe controlled-phaseerrorangpleδ a(cid:12)nd(cid:11)rotationer-
After detuning quickly we therefore obtain ror E depend on the errors in ω and t(cid:12) , respectively.
θ on on
Finally, it is also interesting to consider the fully adi-
2ω +η
11 exp i π b 11 , (30) abatic limit of the Strauch CZ gate. By this we mean
→− − √2g
(cid:20) (cid:18) b (cid:19)(cid:21) that the switching is adiabatic with respect to both gb
(cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12) and η. For the gate time to be competitive with the
or, using expressions (18) and (19),
nonadiabatic gate of Table I, a larger coupling g is re-
b
quired, which might lead to significant higher-order and
11 e−i(α+β) 11 . (31)
→− cross-coupling errors in a multi-qubit device, but in the
The two phase(cid:12) a(cid:11)ngles α and β(cid:12)can(cid:11) be cancelled by fully adiabatic limit only one pulse control parameter—
(cid:12) (cid:12)
eitherω ort —needstobeoptimized(twoz rotations
the application of independent auxiliary single-qubit z on on
are still required). This is because adiabaticity now as-
rotations
sures that the 11 population is preserved (apart from
| i
R (γ) exp[ i(γ/2)σz] (32) exponentiallysmallswitchingerrors),takingovertherole
z
≡ − previouslyplayedbyt ,andasinglepulseshapeparam-
on
tothequbitandbus. Qubitz rotationsareimplemented eterissufficienttospecifythecontrolledphase. Ahighly
by frequency excursions, whereras resonator z rotations adiabatic CZ gate was demonstrated in Ref. [43].
8
There are a few important differences between the the compensating phase shifts are actually implemented
Strauch CZ gate applied to a pair of directly coupled in practice.
qubits (as in Ref. [42]) and to the qubit-bus system con- Thisuseofinteractingsystemeigenfunctionsandcom-
sidered here. These differences result from the harmonic pensatingphaseshiftsasdescribedaboveprovidesacom-
spectrum of the resonator in the latter case and are dis- putational basis that evolves ideally between gates, but
cussed below in Sec. IVC. such an approach is not scalable; for example, there are
22n+1 such computational states in QVN . In Ref. [34]
n
an approximate but scalable implementation of this ap-
C. Eigenstate basis
proachwasintroduced. Theideaisthattheexactenergy
E of a computational state in QVN is, to an extremely
n
The Hamiltonian (7) is written in the usual bare basis goodapproximation,thesumofuncoupledqubitandres-
of uncoupled system eigenstates, but information pro- onator frequencies, i.e., essentially noninteracting. This
cessingitself is best performedin the basis ofinteracting is not simply a consequence of the dispersive regime en-
eigenfunctions of Hidle, where Hidle is given by (7) with ergies (eigenvalues of Hidle), which have non-negligible
the qubits in a dispersive idling configuration [34]. This interactioncorrections,butbecauseonlyaspecialsubset
choice of computational basis assures that idling qubits oftheeigenfunctionsareusedforinformationprocessing:
suffernopopulationchangeinthedecoherence-freelimit, In the QVN system we only make use of H eigen-
n idle
and evolve in phase in a way that can be almost exactly functions in which there are no more than n excitations
compensatedforbyanappropriatechoiceofonlynrotat- present, and such that two directly coupled elements—
ing frames or local clocks, one for each qubit [34]. Here qubits or resonators—are not simultaneously occupied
we briefly review this important concept. (except during the CZ gate). For example, when the
In principal, any complete orthonormal basis of the data is stored in memory, the residual memory-memory
physical Hilbert space that can be appropriately pre- couplingisfourthorderinthequbit-resonatorcouplingg
pared,unitarily transformed,andmeasured—essentially, (for simplicity we assume here that g =g ). This leads
b m
anybasiswhereonecanimplementtheDiVincenzocrite- to an eighth-order conditional frequency shift (order g16
ria[55]—isavalidbasisonwhichtorunaquantumcom- idling error) [34]. Next, suppose an excitation is trans-
putation. Defining the computational states to be inter- ferred from memory to a qubit via a MOVE gate. Now
acting system eigenfunctions gives them the simplifying the dominant frequency shift is sixth order. And when
propertythatthetimeevolutioncanbedecomposedinto anexcitationisinthebusthelargestshiftisfourthorder
asequenceofgates,betweenwhich(almost)noevolution [34]. The largest idling error (associated with the phase
occurs. In other words, idling between gates generates compensation) is therefore eighth order in g and can be
the identity operation. This property, which is implic- made negligible with proper system design.
itly assumed in the standard circuit model of quantum
The compensating phase shifts could be implemented
computation, could be realized in an architecture where
through additional local z rotations, one for each qubit
the Hamiltonian H can be completely switched off be-
and resonator. However, these phase shifts evolve in
tween gates. However, it is not possible to set H = 0 in
time with veryhigh(>1GHz) frequency,andit is there-
the QVN architecture; nor can H itself be made negligi-
fore experimentally more practical to introduce a local
blysmallbetweengates. Therefore,nonstationarystates
clock/rotating frame for each qubit and resonator. This
such as uncoupled-qubit eigenstates accumulate errors
is achieved by introducing a fixed-frequency microwave
(including population oscillations) between gates unless
line for each qubit and resonator, and measuring each
a correction protocol such as dynamical decoupling [56] qubit and resonator phase relative to the phase of its
is used. By defining computational states in terms of
reference. By choosing the frequency of the qubit (res-
interacting system eigenfunctions ψ at some prede-
onator) reference microwave equal to the idle frequency
{| i}
fined dispersive idling configuration (qubit frequencies),
(resonator frequency), the component frequencies [and
the only evolution occuring during an idle from time t
1 therefore the quantity E in (41)] are effectively zeroed,
to t is a pure phase evolution,
2 and no more than 2n+1 different reference frequencies
ψ(t ) ψ(t ) =e−iE(t2−t1) ψ(t ) , (41) or local clocks are required. This procedure corresponds
1 2 1
| i→| i | i to implementing the experiment in a multi-qubit rotat-
where E is the exact energy eigenvalue (and we neglect ing frame. And, in a further simplification, the local
decoherence). Furthermore, it is possible to compensate clocks/rotatingframesforthe resonatorsarereplacedby
for—or effectively remove—the pure phase evolution in additionalqubit z rotations thatare handledin software
(41) by applying phase shifts (after the idle period) to (i.e.,combinedwithfuturerotations). Therefore,inprac-
eacheigenfunctiontocancelthee−iE(t2−t1)phasefactors; tice only n local clocks/rotating frames are needed, one
doingsowouldresultintheidealbetween-gateevolution for each qubit.
Because the CZ gate simulations reported in Table I
ψ(t ) ψ(t ) = ψ(t ) . (42)
1 2 1
| i→| i | i are already supplemented with local z rotations, these
The idling dynamics (42) is evidently equivalent to set- local clocks/rotating frames do not need to be included
ting H = 0 between gates. We will discuss below how in those simulations; we simulate the lab frame. How-
9
ever, they are included in the pulse-shape error simula- in g and g . At the 99.99% fidelity level, it is sufficient
b m
tionsreportedafter(94)andintheCZ qubit-qubitgate to calculate S to first order. Writing H = H +δH
23 idle 0
simulation reported in Sec. IVB. leads to the condition i[S,H ]+δH = 0, which is im-
0
Having motivated the use of interacting system eigen- mediately solvable in the bare basis q q m m b .
1 2 1 2
| ··· ··· i
functions for computational basis states, it is still neces- Here q ,m ,b 0,1,2,... . Other efficient eigenfunc-
i i
∈ { }
sarytoestablishthatsuchstatescanactuallybeprepared tion approximation schemes are also possible.
and measured. Because we can assume the processor to Inthis workwe denote the exactorapproximateH
idle
initially start in its interacting ground state—a compu- eigenfunction perturbatively connected to the bare state
tational basis state—preparation of the other computa- q q q m m m b by
1 2 n 1 2 n
| ··· ··· i
tional states can be viewed as a series of π pulses and
MOVE gates. We expect that such operations on the q q q m m m b , (43)
1 2 n 1 2 n
interacting eigenfunctions can be performed at least as | ··· ··· i
accurately as when applied to bare states. Eigenfunc- following the overline notation introduced above. Note
tion readout is a more subtle (and model-dependent) that(43)isnotatensorproductofsingle-qubit/resonator
question, but the analysis of Ref. [34] suggests that eigenstates as is usually the case.
interacting-eigenfunction readout is actually better than
bare-state readout (in the model considered there).
We also note that the idling configuration and associ-
D. Auxiliary z rotations and CZ equivalence class
ated eigenstate basis generally changes between consec-
utive gates (an example is given below in Sec. IVB). In
Table I, the idling configurationhas qubit q at ω and The standard CZ gate in the bare two-qubit basis
1 off
the others at ω . Therefore, our entangling gate de- 00 , 01 , 10 , 11 is
park {| i | i | i | i}
sign is constrained by the requirement that we start and
end in eigenstates of this particular H . 1 0 0 0
idle
Thediscussionabovemotivatingtheuseofinteracting CZ 0 1 0 0 . (44)
eigenstates is based on their nearly ideal idling dynam- ≡0 0 1 0 
ics. It is still interesting, then, to consider whether the 0 0 0 1
 − 
CZ gate can be generated equally well in either (bare  
or interacting eigenfunction) basis. We find that for the However in the QVN processor, local z rotations can be
parameter regimes considered here, it is not possble to performedquicklyandaccurately,typicallybybriefqubit
achieve better than about 99% fidelity in the bare ba- frequency excursions. Thus, we will consider the limit
sis with the same two-parameter pulse profile (it should where SU(2) operations of the form exp[ i(θ/2)σz]can
−
be possible using more complex pulse shapes). The re- be done on the qubits and bus with negligible error and
maining erroris consistentwiththe size of the perturba- inanegligibleamountoftime(fidelitylossresultingfrom
tive corrections to the bare states in the idling configu- errors in these rotations are discussed in Sec. IIII). We
ration. This exercise emphasizes the importance of per- thereforewanttodefineourentanglinggatemodulothese
forming quantum logic with the system eigenfunctions, z rotations. We will do this by constructing a local-z
which have the built-in protection of adiabiticity against equivalenceclassforanarbitraryelement(gate)inSU(4),
unwanted transitions. and then specialize to the CZ gate.
One might object to the use of interacting eigenfunc- WedefinetwoelementsU andU′ ofSU(4)tobeequiv-
tionsasadesigntool,theexactcalculationofwhichisnot alent, and write U′ ⊜U, if
scalable. However, approximate dispersive-regime eigen-
functions are efficiently computable. A particularly sim- U′ =u Uu , (45)
post pre
ple way to do this is to calculate the generator S of the
diagonalizingtransformationV =e−iS by a powerseries where
1 0 0 0
0 e−iγ2 0 0
u(γ1,γ2)≡Rz(γ1)⊗Rz(γ2)=ei(γ1+γ2)/20 0 e−iγ1 0 , (46)
0 0 0 e−i(γ1+γ2)
 
 
for some rotation angles γ . The local-z equivalence class U corresponding to U is the set of elements u Uu
k post pre
{ }
forall u ,u . ForagivengateU, U typicallyoccupiesa four-dimensionalmanifold,depending onfourrotation
pre post
{ }
10
angles. But because (44) is diagonal, CZ instead forms a two-dimensional sheet,
{ }
1 0 0 0
0 e−iγ2 0 0
{CZ}=phase factor×0 0 e−iγ1 0 . (47)
0 0 0 e−i(γ1+γ2)
 − 
 
The CZgate(44)canbe obtainedbyreachinganypointinthe CZ plane andthenperformingauxiliaryz rotations.
{ }
And it is straightforwardto confirm that [43]
1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
−
0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0
 0 0 1 0⊜0 −0 1 0⊜0 0 1 0⊜0 0 1 0 . (48)
−
0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
       − 
       
We note that bus rotations, which cannot be directly U = CZ [see (44)]. Note that the projected U is
target
implemented with microwave pulses or frequency excur- not necessarily unitary here, and that the first term in
sions, are compiled into future qubit rotations. (50) characterizes the possible leakage from the compu-
The discussion above assumed a pair of qubits or a tationalbasis(non-unitarity)whereasthe secondtermis
qubit and resonator, but it applies to a QVN processor proportional to the square of the Hilbert-Schmidt inner
in the interacting eigenfunction basis (43) after a minor productofU withU . AlthoughU isnotassumedto
target
modification. In the bare basis, the CZ gate is typically be unitary, the expression(50) assumes a pure state and
defined through its action (44) on a pair of qubits (or a is (obviously) not valid in the presence of decoherence.
qubit and resonator). Then, action on a bare computa- [Theformula(50)assumesthattheKrausrepresentation
tional basis state such as q q q m m m b fol- forthecompletelypositiveprocessisnotnecessarilytrace
1 2 n 1 2 n
| ··· ··· i
lows from the tensor-productform of that bare state. In preserving,butithasonlyoneterm.] Theform(50)also
the eigenstatebasisthe CZgatemustbedefinedthrough assumes an average over a 4-dimensional Hilbert space;
its action on in the N-dimensional generalization the denominator is
N +N2, which is necessary (note numerator) to assure
|q1q2···qnm1m2···mnbi, (49) that Fave(Utarget,Utarget) is unity.
Itisalsousefultocalculatetheminimumorworst-case
such as to reproduce the ideal action on the bare states
fidelity. Theminimumfidelityofinteresthereisthestate
towhichtheyareperturbativeyconnected. Forexample,
fidelity minimized over initial computational states, for
the CZ gate on qubit q and the bus acts ideally as
1 a gate that has already been optimized (by maximizing
F ). InSec.IIIBwearguedthatthedominantintrinsic
CZ 0q q q m m m m 0 = 0q q q m m m m 0 ave
2 3 4 1 2 3 4 2 3 4 1 2 3 4 error mechanism (for an optimal pulse) in the truncated
CZ(cid:12)(cid:12)0q2q3q4m1m2m3m41(cid:11)=(cid:12)(cid:12)0q2q3q4m1m2m3m41(cid:11) qubit-resonator model (9) is the nonadiabatic excitation
of the 02 channel, in which there are two photons left
CZ(cid:12)(cid:12)1q2q3q4m1m2m3m40(cid:11)=(cid:12)(cid:12)1q2q3q4m1m2m3m40(cid:11) in the |busi resonator. Therefore, in the model (9), the
CZ(cid:12)(cid:12)1q2q3q4m1m2m3m41(cid:11)=−(cid:12)(cid:12) 1q2q3q4m1m2m3m4(cid:11)1 , minimum state fidelity occurs for the initial eigenstate
11 . In the QVN processor, this worst-case state is
4
wher(cid:12)(cid:12)e qi,mi ∈{0,1,2,...}(cid:11). (cid:12)(cid:12) (cid:11) w| riitten [in the notation of (43)] as
100000001 , (51)
E. Fidelity definitions | i
where we have assumed a CZ gate between qubit q and
1
Thegateorprocessfidelitymeasureweuseinthiswork the bus. Numerical simulation of this gate in the QVN4
isbasedonastatefidelitydefinedbytheinnerproductof processor confirms that the minimum CZ fidelity indeed
the ideal and realized final (pure) states, squared. This occurs for the initial state (51), and is due to leakage
leads to a state-averagedfidelity given by [57, 58] from the computational subspace. We therefore define
the minimum fidelity to be the state fidelity for initial
Tr(U†U)+ Tr(U† U) 2 condition (51),
target
F (U,U ) , (50)
ave target
≡ 20
(cid:12) (cid:12) 2
(cid:12) (cid:12) F|11i 100000001U 100000001 . (52)
whereU istherealizedtime-evolutionoperatorinthein- ≡ h | | i
teracting eigenfunction basis after auxiliary z rotations, Note that this exp(cid:12)ression is not sensitive to(cid:12)the value of
(cid:12) (cid:12)
projected into the relevant computational subspace, and the controlledphase, and only accounts for leakage from