Scalable quantum computation architecture using always-on Ising interactions via quantum feedforward

1.3 MB·English

by  Takahiko Satoh

Scalable quantum computation architecture using always-on Ising interactions via
quantum feedforward
Takahiko Satoh,1,2,∗ Yuichiro Matsuzaki,1 Kosuke Kakuyanagi,1
William J. Munro,1 Koichi Semba,3 Hiroshi Yamaguchi,1 and Shiro Saito1
1NTT Basic Research Laboratories, 3-1, Morinosato Wakamiya Atsugi-city, Kanagawa 243-0198 Japan
2Department of Computer Science, Graduate School of Information Science and Technology,
The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, Japan
3Advanced ICT Research Institute, National Institute of Information and Communications Technology,
4-2-1, Nukuikitamachi, Koganei-city, Tokyo 184-8795 Japan
(Dated: February 2, 2015)
5 Here,weproposeawaytocontroltheinteractionbetweenqubitswithalways-onIsinginteraction.
1 Unlike the standard method to change the interaction strength with unitary operations, we fully
0 makeuseofnon-unitarypropertiesofprojectivemeasurementssothatwecaneffectivelyturnon/off
2
the interaction via feedforward. We then show how to generate a two or three-dimensional cluster

n state that are universal resource for fault tolerant quantum computation with this scheme. Our
a schemeprovides an alternative way torealize a scalable quantumprocessor.
J

0
3 I. INTRODUCTION Oneofthewaystoreducetherequiredleveloftechnol-
ogyis to use a system with always-oninteraction. There

] are a couple of theoretical proposals for this direction.
h Quantum computation is a new paradigm of informa-
Zhou et al suggested a system with always-on Heisen-
p tion processing. Known algorithms give superior per-
- formances for tasks such as factoring [1, 2], searching an berg interaction [24]. They use interactionfree subspace
t to protectthe targetencodedqubit from the residualin-
n unsorteddatabase[3,4],quantumsimulation[5,6],other
a algorithms[7–12]andmore. Allthesealgorithmsrequire teraction, and they show that only local manipulations
u on the system actually provide universal quantum com-
alargescalequantumcomputer. Aquantumcomputeris
q putation. Simon et al also suggested to use always-on
composedofasequenceofimplementationofsingle-qubit
[
Heisenberginteractionsystemforscalablequantumcom-
gates and two-qubit gates [13–16]. The single-qubit gate

1 denotesarotationofthequbitaroundarbitraryaxisand putationbycollectivelytuningthequbits[25,26]. These
v approachlook attractive due to its simplicity that could
degree. A control-phase gate is one of the typical exam-
2 reduce potential decoherence from the interaction.
ples of two-qubit gates. This gate flips the phase of the
1
target-qubit if and only if the state of the control-qubit Here, we propose a novel way to perform universal
7
7 is 1 . The role of control and target qubit are reversible quantum computation with a system having an always-
| i
0 for control-phase gate. Individual qubits should be effi- on Ising interaction. In quantum mechanics, there are
. cientlyaddressedandtheinteractionbetweentwo-qubits two type of operations, unitary operations such as ap-
1
0 should be controlled by some external apparatus. plyingmicrowavepulsesandnon-unitaryoperationssuch
5 The challenge is how to design and build a quantum as readout of the qubit. While most of the authors in
1 computerwitharealistictechnology. Thisrequiresquan- previous papers use unitary operation to control the in-
: tumarchitecture. Therehavebeenanumberofthesefor teraction [24–26], we exploit the non-unitary properties
v
i relevant physical systems, such as nitrogen-vacancy cen- that the projective measurement have. We will assume
X
tre [17, 18], ion traps [19], superconducting systems [20]. an always-on Ising interaction between nearest neighbor
r Many of those have assumed isolating system and excel- qubits, and will insert an ancillary qubit between the
a
lent controllability. However, in realistic circumstances, qubits that processquantum information. We show that
turning on/off the interaction in a reliable way is one it is possible to effectively turn on/off the interaction
of the hardest parts in such architectures. For example, viaquantummeasurementandfeedforwardonthe ancil-
two-qubit gates require in-situ turn on/off the interac- lary qubits. Furthermore, we explain how scalability is
tion between qubits by the external control apparatus. achievedinthisscheme,andsuggestawaytoconstructa
Since imperfection of the interactioncontroltends to in- largetwoorthree-dimensionalclusterstatewhichenables
ducecorrelatederrorsbetweenqubits,sophisticatedtech- us to perform fault tolerant quantum computation with
nology is required to suppress such error rate below the high error threshold [23]. Since quantum feedforward
thresholdoffaulttolerantquantumcomputation[21–23]. technology is becoming matured technology [27–36], our
However, varying the interaction between qubits in-situ proposal provides a feasible and reliable way to control
is not possible for all physical systems. the interaction,whichisa crucialstepforthe realization
of quantum information processing.
Theremainderofthispaperisorganizedasfollows. In
Sec. II, we review the preliminaries of this paper. Sec-
∗ [email protected] tion III presents the detail of our scheme to show how
2
always-interactionis effectively turned on/offvia projec- withahighfidelity. Onlyafewexperimentshavedemon-
tive measurement to ancillary qubits and quantum feed- strated a high fidelity controllable two-qubit gate with
forward. In Sec. IV and Sec. V, we propose the way to a fidelity above the threshold of fault tolerant quantum
generate two and three-dimensional cluster states using computation [74–76]. One of the possible ways to over-
qubits arranged on a plane. Section VI concludes our come the experimental difficulties for demonstrating the
discussion. high-fidelitytwo-qubitgatesistouseanalways-oninter-
actionscheme[24–26,77–79]. Sincetherearenoneedfor
the additional controlling operations to switch the inter-
II. GRAPH STATES AS A RESOURCE FOR action,theseschememayscalewellforalargenumberof
QUANTUM COMPUTATION qubits. Here, we propose a new approach to implement
fault tolerant quantum computation with always-on in-
Let us review the concept of a graph state introduced teraction by using the non-unitary properties of projec-
in [37–40]. A graph G(V,E) is composed of vertices V tive operations and quantum feedforward.
and edges E where an edge connects two vertices. By
regarding the vertex as a qubit and edge as an entangle-
mentbetweenthequbits,wecanassociatethegraphwith III. EFFECTIVE INTERACTION CONTROL
VIA PROJECTIVE MEASUREMENTS AND
agraphstate Φ definedasthe followingequation
| iG(V,E) QUANTUM FEEDFORWARD
Φ = Uˆ(a,b) + ⊗N (1)
| iG(V,E) CZ | i A. Interaction switching with quantum feedforward
(a,Yb)∈E
where (a,b) E denotes an edge between the vertices We introduce the Hamiltonian to realize our scheme
∈
a and b. Also, Uˆ(a,b) denotes a controlled-phase (π) op- to turn on/off the interaction effectively via projective
CZ
eration between them. It is worth mentioning that, a measurements and quantum feedforward. The physical
specific type of a graph state such as a two or three- device that we consider is a general solid-state system
dimensional cluster state can be a universal resource for where every qubit can be individually controlled by a
measurement-basedquantumcomputation(MBQC)[41– microwave pulse and there are always-oninteraction be-
44] and topological quantum computation [21–23]. The tween nearest neighbor qubits. Throughout of this pa-
controlled-phasegatecanbe realizedbyIsingtype inter- per, we assume the following Hamiltoniancorresponding
action[41,42]. WhenagraphG(V,E)isgiven,theneces- to G(V,E)
sary Hamiltonian to create a cluster state corresponding
Hˆ =Hˆsystem+Hˆinteraction
G is as follows G(V,E) G(V,E)
ω
Hˆ = g 1+Zˆl1+Zˆl′ (2) HˆGsy(sVt,eEm) = 2lZˆl+λl(t)cos(ωl′t+θ)Xˆl (5)
G(V,E) (l,l′) 2 2 Xl∈V (cid:16) (cid:17)
(l,Xl′)∈E Hˆinteraction = g(l,l′)ZˆZˆ (6)
where g denotes the interaction strength between G(V,E) 4 l l′
qubit l a(ln,ld′) l′. By letting a separable state + (l,Xl′)∈E
l∈V l
evolve for g t =π according to this Hamilto⊗nian,|thie where ω, λ(t), ω′, and θ denote the qubit energy, Rabi
(l,l′)
following unitary operator will be applied to the initial frequency, microwave frequency, and a phase of the mi-
state crowave. Inmostofthesolid-statesystems,itis possible
to control the value of λ(t) by changing the power of
1+Zˆ 1+Zˆ microwave with much higher accuracy than the case of
l l′
SG(V,E) =exp−iπ 2 2  (3) two-qubitgates. Wemovetoarotatingframedefinedby
(l,Xl′)∈E
=  UC(lZ,l′).  (4) Uˆ =exp −i N ω2l′Zˆlt! (7)
(l,Yl′)∈E Xl
where ω′ denotes its angular frequency of the rotating
and hence we can create the target graph state. l
frameatthesitel,andusearotatingwaveapproximation
Although there are many proposal to realize Ising
so that we could obtain the following Hamiltonian
type interaction such as ultracold atoms in an opti-
cchalarlgaettqicuebi[t45[–5592],],suiopnerctroanpdsuc[t5i3n–g58s]p,insuqpuebrciton[6d0u]c,tisnug- Hˆ ≃ ωl−2 ωl′Zˆl+ λl2(t)Aˆθl + g(l4,l′)ZˆlZˆ(l′8)
perconducting flux qubit [61], resonator waveguide [62], Xl∈V (cid:18) (cid:19) (l,Xl′)∈E
nitrogen-vacancycenter[17,63–68],quantumdot[69–72]
where
andelectronicspinscoupledtothemotionofmagnetized
mechanical resonators [73], the major challenge for ex- 0 e−iθ
Aˆθ = . (9)
perimentalrealizationistoswitchon/offtheinteraction eiθ 0
(cid:18) (cid:19)
3
Unless when required to perform single qubit gates, we
turn off the microwave and set λ = 0, and therefore the
Hamiltonian introduced here is effectively the same as
an Ising model with always-oninteraction. On the other
hand,fortheimplementationofaccuratesingle-qubitro-
tations, we assume a large Rabi frequency as λ g
≫
so that the coupling strength from the nearest neighbor
qubit can be negligible. We will later discuss an error
accumulation due to imperfect single qubit rotation in
detail.
The Hamiltonian described above has an interesting FIG. 1. Schematic of our scheme to implement two-qubit
property that an interaction from other qubits can be gates via projective measurements and quantumfeedforward
turned off by preparing the state of a qubit in a ground udnertheeffect of always-on Ising interaction. Welet evolve
state. To explain this, we consider a specific qubit A thestate|φiAC⊗|+iB accordingtotheHamiltonian,perform
and other qubits interacting with the qubit A, and the a projective measurement onto the middle qubit, and rotate
Hamiltonian of those is described as follows. themiddle qubit back into a ground state so that a C-Phase
can be implemented between the qubit A and C. Due to the
Hˆ′= ωj−ωj′Zˆ+g(A,j)Zˆ Zˆ +ωA−ωA′ Zˆ (10) ethnagtinteheereqdubHiatmAiltaonndiaCndfoorems ntohtatevwoelvemaankye,mtohriesegvueanraunntdeeesr
A  2 j 4 A j 2 A theeffect of the always-on Ising typeHamiltonian.
(AX,j)∈E
= g(A,j)1+2ZˆA1+2Zˆj+ωj−ωj′2−g(A2,j)Zˆj (11) for a time t= π, and perform Yˆ basis
! g
(AX,j)∈E
1
with ( 1 = (0 i1 ) (16)
|± Yˆi √2 | i± | i
1
ωA−ωA′ = 2g(A,j). (12) measurement on the middle qubit B. The state after the
(AX,j)∈E measurement is written as
Interestingly, if the qubit A is prepared in a ground ρ′ =Pˆ±e−iHˆtρeiHˆtPˆ± (17)
± B B
state, the interaction from the qubit A cancels out be-
cause of where denotes the measurement result. Here,
±
1+ZˆA1+Zˆj Pˆ± = 1 1 Yˆ (18)
g 0 =0. (13)
(A,j) 2 2 | iA 2 ±
(cid:16) (cid:17)
denotes a projection operator on the qubit B. Finally,
This means that preparing a specific qubit in a ground
we perform a quantum feedforwardoperation, that is an
state effectively turn off the interaction between this
implementation of different local operations depending
qubit and nearest-neighbor qubit. Therefore, if all
onthemeasurementresults,ontothequbitBsothatthe
nearest-neighborqubitsaregroundstate,thequbitisnot
qubit B can be prepared in a ground state. We define a
effectedbyanyinteractions,whichisthe strikingfeature
feedforward operator as
of our scheme.
Althoughwewilllaterdiscussthecaseoftwoorthree- Fˆ± =Sˆ±Uˆ∓2π,XˆSˆ± (19)
dimensional cluster state that is a universal resource for ABC A B C
quantum computation [23] , we start by explaining how where Sˆ± denotes a shift gate defined as
to generate a one-dimensional three-qubit cluster state
(Fig. 1) to illustrate our concept about how to control 1 0
(20)
theeffectiveinteractionviaprojectivemeasurementsand 0 i
quantum feedforward. Here, we assume all interaction (cid:18) ± (cid:19)
strength are equally g. Firstly, we prepare a separable and Uˆθ,Xˆ denotes a single-qubit rotating around x-axis
+ state for the qubit B, and prepare an arbitrary pure rotationwithanangleofθ. Thestateafterthe quantum
| i
state for the qubit A and C. An initial state is described feedforward is described as
by
ρ =Fˆ+ Pˆ+ρPˆ+Fˆ+† +Fˆ− Pˆ−ρPˆ−Fˆ−† (21)
final ABC B B ABC ABC B B ABC
ρ= φ φ + + . (14)
| ih |AC ⊗| ih |B =UˆC(AZ,C)|φihφ|ACUˆC(AZ,C)⊗|↓ih↓|B. (22)
Secondly, we let the system evolves by the Hamiltonian
Therefore, after these operations, controlled-phase oper-
Hˆ= g 1+ZˆB1+Zˆj+ωj−ωj′−g(B2,j) Zˆ (15) athtieonstsaateredpoeesrfonromteedvoblveetwaeneynmtohreeqbuebcaitusAe tahnedqCu,biatnBd
(B,j) 2 2 2 j
! is prepared in a ground state.
j∈X(A,C)
4
A B C
g g We let the state evolve for a time t , perform π pulse to
1 + 2 ψ = φ ⊗ + qubit A, andlet the state evolvefor1a time t . The total
AC B 2
Echo pulse t unitary evolution can be described by
1
π(g +g ) π(g −g )
t1= 2g1 g 2 ,t2 = 2g1 g 2 Uˆ =exp ig (t t )1+ZˆA1+ZˆB
t 1 2 1 2 − 1 1− 2 2 2
2
ψ =Uˆ(B,C)Uˆ(A,B) φ ⊗ + (cid:16) 1+ZˆB 1+ZˆC
±Yˆ CZ CZ AC B −ig2(t1+t2) 2 2 , (27)
Y measurement and quantum feedback
(cid:17)
0 ψ =Uˆ(A,C) φ ⊗ 0 and so,
CZ AC B
Uˆ φ =Uˆ(B,C)Uˆ(A,B) φ (28)
| iABC CZ CZ | iABC
FIG. 2. Controlled-phase operations using a spin echo tech-
niquewithasymmetriccouplingstrength. Uˆ =Uˆ(A,B)Uˆ(B,C) sothatwecanperformcontrolled-phasegatesevenifthe
CZ CZ
isperformedontheinitialstateataspecifictimingduetothe couplingstrengthisasymmetric. Afterthisevolution,we
implementationofaπpulsewhereUˆ(j,k)denotesacontrolled- use projective measurements and quantum feedforward
CZ
phase operation between qubit j and k. to effectively turn off the interaction as long as qubit B
is prepared in a ground state. Therefore, we succeed in
performing controlled-phase operation between qubit A
Meanwhile,ifwesetthequbitBinanexcitedstateby
and C.
quantum feedforward operation, the final state become
as follows.
ρ′ =e−iHˆt′ Uˆ(A,C) φ φ Uˆ(A,C) eiHˆt′(23) C. One dimensional cluster state
final CZ | ih |AC CZ ⊗|↑ih↑|B
(cid:16) (cid:17)
where t′ denotes an arbitraly short-time. Interactions
Weonlyexplainaboutthree-qubitcasehere. However,
cause the extra phase shift between (or )
AB BC itisstraightforwardtogeneralizethisideatoanarbitrary
|↑↑i |↑↑i
and other three states of qubit A and B (or B and C). It
size of one-dimensional cluster state, because we can ig-
is considered that these extra phases cause operational
norethe coupling fromthe otherqubits aslongas wein-
errors even if we can appropriately correct these phases.
sertanancillaryqubitpreparedinangroundstateaswe
Additionally, an excited state can be quenched suddenly
discussed before. It is worth mentioning that the necce-
to a ground state. For these reasons, we set the qubit B sary number of the π pulses increases linearly against
in a ground state after the projective measurement.
the number of the qubits, due to the use of such ancil-
laryqubits to stopthe interactionfromthe otherqubits.
Additionally, our scheme can be applied to two or three-
B. Asymmetric coupling strength case
dimensional cluster state. In these cases, we repeatedly
implement the similar procedureas we use in the case of
Interestingly, our scheme to perform controllable a one-dimensional cluster state so that we can balance
controlled-phase gates under the effect of always-on in- theinteractionsjustbyaddingafewoperations. We dis-
teractionworksevenwhenthecouplingstrengthbetween cussthedetailsofthoseinteraction-balancingschemesin
qubits is different. We adopt a spin echo technique [80– AppendixA.Hence,throughoutofthispaper,weassume
82]tobalancetheinteraction. Inthespinechotechnique, that all interactions are equal.
implementation of a π pulse can refocus the dynamics
of the spin so that the effect of the interaction should
be cancelled out. Suppose that we have three qubits in
D. Unavoidable error of feedforward operation
a raw, and the coupling strengths between the nearest
neighbor qubits are g and g as shown in Fig. 2 where
1 2
we assume g > g without loss of generality. Here, we Itisworthmentioningthatwecouldnotavoidadetun-
1 2
introduce ingerrortoperformasinglequbitrotationinouralways-
on interaction system. In a solid-state system, it is typ-
π(g +g )
t = 1 2 (24) ically possible to perform a high-fidelity single qubit ro-
1 2g1g2 tation by applying a on-resonantmicrowavepulse whose
frequency is the same as the qubit energy. However, in
and
our case, the target qubit has an unknown energy shift
π(g g )
t = 1− 2 (25) due to the interaction when a state of nearest neighbor
2 2g1g2 qubitscontainsasuperposition. Asanexample,weagain
consideracaseofthree-qubitone-dimensionalchain,and
to satisfy
estimate the fidelity to perform a π pulse on the middle
2
g (t t )=g (t +t )=π. (26) qubit prepared in a ground state, and the Hamiltonian
1 1 2 2 1 2
−
5
denotes the duration of the microwave π pulse. This
TABLE I. The effective Hamiltonian of qubit B depends on 2
means that, by increasing the Rabi frequency λ , we
thestatesofqubitAandCinFig.1. Whenthestatesofthe B
qubitAandCcontainssuperposition,theresonantfrequency can suppress this detuning error. We plot this error ǫπ2
against the coupling strength g and the Rabi frequency
of the qubit B is not uniquely determined. Since the mi-
crowave frequency is fixed, ZˆB component induce the detun- λB in Fig. 3.
ingerrorǫπ whenwerotatequbitB.Asthetableshows, the
2
detuningerror ǫπ becomes maximum for the case of |↑↑iAC.
2
QubitA and C (state) QubitB (effective Hamiltonian)
|↑↑iAC HB =gZˆB+ λ2BYˆB
|↑↓iAC, |↓↑iAC HB = g2ZˆB+ λ2BYˆB
|↓↓iAC HB = λ2BYˆB
of this system is described as follows.
ω ω′ λ (t)
Hˆ l− lZˆ + B Aˆθ
ABC ≃ 2 l 2 B
l∈A,B,C
X
g
+ (l,l′)ZˆZˆ . (29)
l l′
4
(l,l′)∈(AX,B),(B,C)
Here, we set mixing angle
FIG. 3. The worst rotating error (ǫπ) and the interaction
π 2
θ = (30) strength (gT2) between each pair of nearest neighbor qubits
2 in switching scheme (Fig. 1) against various Rabi frequency
and microwave frequency (λBT2). Here, T2 denotes thecoherence time of the qubit.
ω′ =ω g (31)
B B − Throughoutofthispaper,whenwecalculateafidelity,
to obtain the following Hamiltonian. we always consider the worst case for detuning error as
discussedabove. The effective Hamiltonian of the target
1 g λ
Hˆ′ = ω ω′ Zˆ + BYˆ qubit to be rotated by the microwave is described by
ABC 2 l− l− 2 l 2 B
l∈XA,C (cid:16) (cid:17) H = λXˆ (35)
1+Zˆl1+Zˆl′ target 2 target
+ g . (32)
2 2
when all nearest neighbor qubits are in a ground state
(l,l′)∈(AX,B),(B,C)
while the worst case of the Hamiltonian is
We show the effective Hamiltonianof the qubit B of this
g λ
caseinTableI.Sincetheresonantfrequencyofthequbit H = nZˆ + Xˆ (36)
target target target
B depends onthe state ofthe qubitAandC,it becomes 2 2
impossible to apply on-resonant pulse on the qubit B if when all nearest neighbor qubit is in an excited state,
one of these states have a superposition. To implement and we fix the frequency of
our scheme to control the interaction, we have already
g
cphuolsseenanthdetfhreeqwuoenrsctyfiodfe(liωtyB (−whg)enfotrhethaectmuiaclroewffaevcteivπ2e ωtarget− 2n (37)
Hamiltonian of the qubit B is for the microwave π pulse. Here, ω is the original
2 target
resonantfrequency ofthe targetqubitandndenotes the
λ
HB =gZˆB+ BYˆB) (33) number of qubits interactingwith the targetqubit. This
2
willenable us to evaluatethe performanceofour scheme
can be calculated as follows. for the fault tolerant quantum computation.
2
ǫπ=1 e−iHˆBt +
2 − h↓| | i
(cid:12)(cid:12)cos(1t 4g2+(cid:12)(cid:12)λ2)+sin(1t 4g2+λ2) 2gi+λB 2 E. Optimal interaction strength
(cid:12) 2 (cid:12)B 2 B √4g2+λ2
=1 B
−(cid:12) p √2 p (cid:12) Here, we discuss the optimal interaction strength be-
(cid:12) (cid:12)
(cid:12) (cid:12) tween the qubits to perform a high fidelity controlled-
(cid:12) (cid:12)
where (cid:12) (cid:12) phase gate. Since the coherenece of the quantum states
(cid:12) (cid:12)
π degrades due to decoherence, we need to perform a
t= (34) controlled-phase operation much faster time scale than
2λ
6
the coherence time of the qubit. For this purpose, we
need to increase the coupling strength to realize a fast (cid:1)(cid:0)(cid:2)(cid:2)
controlled-phase gate. However, the strong coupling
strength makes it difficult to perform an accurate quan- (cid:1)(cid:2)(cid:2)(cid:2)
tum feedforward operations because the always-on cou- (cid:2)(cid:3)(cid:8)(cid:8)(cid:5)
pling between qubits induces unknown energy detuning
(cid:4)(cid:0)(cid:2)(cid:2)
of the qubit frequency as described before. So there (cid:12)(cid:13)(cid:11) (cid:2)(cid:3)(cid:8)(cid:8)(cid:4) (cid:10)
should existan optimal interactionstrengthto minimize
(cid:4)(cid:2)(cid:2)(cid:2)
the controlled-phase gate error that comes from the de-
(cid:2)(cid:3)(cid:8)(cid:9)(cid:8)
coherence of the qubits and imperfect quantum feedfor-
ward operations. Decoherence error that we consider is (cid:0)(cid:2)(cid:2)
general Markoviannoise. We assume that the error rate (cid:2)(cid:3)(cid:8)(cid:9)(cid:7)
increases exponentially against time as
(cid:5)(cid:2)(cid:2)(cid:2)(cid:2) (cid:17)(cid:2)(cid:2)(cid:2)(cid:2) (cid:0)(cid:2)(cid:2)(cid:2)(cid:2) (cid:6)(cid:2)(cid:2)(cid:2)(cid:2) (cid:7)(cid:2)(cid:2)(cid:2)(cid:2) (cid:9)(cid:2)(cid:2)(cid:2)(cid:2)
ǫd = 12(1−e−TTC2Z). (38) (cid:16)(cid:15)(cid:14)
(a)
Here, T denotes the coherence time of the qubit and
2
π
T (= ) (39)
CZ
g
denotes the gate operation time. Since we consider a
parameter regime for T T , we can simplify the
CZ 2
≪
decoherence error as
T
CZ
ǫ = . (40)
d
T
2
Weassumethatthesingle-qubitoperationscanbeimple-
mented much faster than the coherence time, and hence
the decoherenceeffect during the single qubit operations
is negligible compared with other effect such as deco-
herence during the controlled-phase gate. The setup
we consider for the estimation of the optimal coupling
(b)
strength is as follows. As described in Fig. 1, to per-
form a controlled-phase gate, we use two main qubits A
andCandoneancillaryqubitBthatisinsertedbetween FIG. 4. (a) An achievable fidelity (F) and the optimal cou-
plingstrength(gT )of acontrolled-phasegate inourscheme
the main qubits. Initially, an ancillary is prepared in a 2
(Fig.1)against variousRabifrequency(λT ). Thesolid line
ground state and main qubit are prepared in arbitrary 2
denotes F and the dashed line denotes gT . (b) An achiev-
states. Also, for simplicity, we assume that all nearest- 2
able fidelity (F) against interaction strength (gT ) and Rabi
2
neighbor coupling strength between qubits are equal be-
frequency (λT ).
2
tween these three qubits. We evaluate the achievable
fidelity during the implementation of a controlled-phase
gate in our scheme (Fig. 1). Firstly, by performing π
2
pulse, we rotate the ancillary qubit B from ground state
rangeofthe Rabifrequency λ in Fig.4(a). Also,we plot
into + state. At this time, the qubit B have unknown
| i the relationship between an achievable fidelity, the opti-
energy shift due to the coupling from qubit A and C
malinteractionstrengthandRabifrequencyinFig.4(b).
during the rotationsothat a detuning erroroccurs. Sec-
Thisshowsthatanachievablefidelity(F)monotonically
ondly, let evolve the system according to the Hamilto-
increase with the increasing Rabi frequency (λ) and in-
nian. During this time evolution, every qubit is affected
teraction strength g has the optimal point against λ.
by environmental noise, and so decoherence error accu-
mulates. Finally, we measure Yˆ and perform quantum In this paper, we do not discuss about the details of
feedforward on the qubit B. Again, due to the coupling theerrorsinprojectivemeasurementsandquantumfeed-
from nearest neighbor qubits, qubit A and C suffers the forward operations. But we can treat these errors as a
detuning error for the feedforward rotations while the type of additional dephasing error. For example, in our
qubit B can be accurately rotated by a resonant mi- switching schemeofIIIA,we assume thatwe fail to per-
crowave pulse. Therefore, the achievable fidelity is cal- formmeasurementorfeedforwardoperationwithaprob-
culated as F = 1−(2ǫπ2 +3ǫd), where we assume that ability of ǫm. The ancillary qubit B become
the error makes the state orthogonal to the ideal one
to consider the worst case. We plot the achievable fi-
delity F and interaction strength g corresponding to the ρ =(1 ǫ ) +ǫ . (41)
B m B m B
− |↓ih↓| |↑ih↑|
7
At this time, the state of the total system is written as thatcontainsquantuminformation. Theotherqubitsare
called syndrome qubits. Half of syndrome qubits are to
ρABC=(1−ǫm)UˆC(AZ,C)|φihφ|ACUˆC(AZ,C)⊗|↓ih↓|B detectthedephasingerrorsonthelogicalqubitswhilethe
+ǫ e−iHˆtUˆ(A,C)φ φ Uˆ(A,C) eiHˆt(42) other ancillary qubits are to detect bit-flip noises. Main
m CZ | ih |AC CZ ⊗|↑ih↑|B qubits are set on a grid point of square lattice and an-
where cillary qubits for effectively turning on/off interactions
are set on the midpoint of these main qubits. There
Hˆ = g1+Zˆl1+Zˆl′ (43) are Ising type interactions between each pair of nearest
2 2 neighbor main qubit and ancillary qubit. To generate
(l,l′)∈(AX,B),(B,C) a whole two-dimensional cluster state, we need to per-
denotes Ising type interactions between qubit A, B and form controlled-phase operations between every pair of
B,C. BytraceoutoftheancillaryqubitB,wecantreat the nearest logical and syndrome qubits as described in
the effects of the interactions Hˆ as dephasing errors at the following procedure. Firstly, we prepare all ancillary
most ǫ σ on the qubit A and C. Such error can be qubits between main and syndrome qubit to + state.
correctmedZby the quantumerrorcorrectionas long as the Secondly, we let the states evolve for a time t|=iπg. Fi-
error is less than the threshold. nally,weperformprojectivemeasurementsandquantum
feedforward operations to all ancillary qubits for gener-
ating two-dimensionalcluster state. In these operations,
IV. GENERATION OF A TWO DIMENSIONAL each ancillary qubit has no effect on the state of other
CLUSTER STATE UNDER THE EFFECT OF ancillary qubits, so that we can handle the effect of each
ALWAYS-ON INTERACTION FOR SURFACE operation as individual three-qubits system and we can
CODING SCHEME proceed all controlled-phase operations simultaneously.
Furthermore, similar to the case of a one-dimensional
In this section, we show how to apply our scheme to cluster state, the energies of ancillary qubits have un-
generate a two-dimensional cluster state, which is a uni- known energy shifts as described in Table I so that the
versal resource [21, 22] for quantum computation. We upper bound fidelities and optimal interaction strengths
of each controlled-phase operations coincide with three-
qubitcaseshowninFig.4(a). Sincescalablesurfacecod-
)& *(cid:29)"+(cid:28)’(cid:25)(cid:30)!(cid:27) ((cid:19)! ’(cid:25)(cid:30)!(cid:27)
ing scheme requirethe errorrate aroundbelow 1 %, this
result shows that the Rabi frequency should be tens of
thousands times larger than the decay rate ( 1 ) and the
T2
coupling strength should be thousands times largerthan
that.
% (cid:23)!(cid:24)(cid:24)(cid:19)(cid:29)&’(cid:25)(cid:30)!(cid:27)
V. GENERATION OF A THREE
DIMENSIONAL CLUSTER STATE UNDER THE
(cid:18)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:26)(cid:27)(cid:19)(cid:27)(cid:28) (cid:18)(cid:30)(cid:20)(cid:31) !(cid:27)(cid:23)(cid:28)(cid:24)(cid:24)"#$(cid:29)"(cid:23)(cid:28)(cid:26)(cid:26)"(cid:29)
EFFECT OF ALWAYS-ON INTERACTION FOR
TOPOLOGICAL QUANTUM COMPUTATION
Although we discussed how to generate a two-
dimensionalclusterstateabove,wecanapplyourscheme
to generate a three-dimensional cluster state, which is a
(cid:18)(cid:23)(cid:20),-(cid:28)$(cid:29)"(cid:23)(cid:28)*(cid:25)(cid:29)(cid:28)#"(cid:29)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:26)(cid:27)(cid:19)(cid:27)(cid:28).(cid:28) (cid:28)(cid:29)(cid:19)(cid:27)!"
universal resource for the topological quantum compu-
FIG. 5. (a)Schematic to generate a two-dimensional cluster tation [23]. Topological quantum computation is known
statewithalways-oninteractionusingourscheme. Hollowcir-
to have a high threshold for quantum error correction
clesdenotelogicalqubitsandcoloredcirclesdenotetwotypes
especially when there is a finite probability to lose a
ofsyndromequbitsforsurfacecodingscheme. Logical qubits
hold arithmetical quantum information. Blue qubits deny Xˆ qubit [83, 84]. The overview of this scheme is shown
errors,andredqubitsdenyZˆ errors. Werefertothesequbits inFig.6 where three-dimensionalcluster state is usedas
asmainqubits. (b)Aunitcelltogenerateatwo-dimensional a resource for the computation. In the 3D cluster state,
cluster state. The large circles denote main qubits while the qubits connected in a z-axis direction are used for the
smallcirclesdenoteancillaryqubitsforswitchinginteractions. logical qubit that contains the information for the com-
TheedgebetweenthequbitsdenotetheIsingtypeinteraction putation (Fig.2). The other qubits located between the
between those qubits. logical qubits are used for detecting error syndrome. In
order to process the computation with error corrections,
now give the overviewof our setting in Fig. 5(a) and the we measure qubits by layer. Syndrome qubits are mea-
physical implementation in Fig. 5(b). There are three suredinXˆ-basis,andtheoutcomesareusedfordetecting
typesofmainqubits. Oneofthemiscalledalogicalqubit the location and type of the error so that we can correct
8
theerrorafteranalyzingthesyndromesbyclassicalcom-
dYUMQQHTReKWMV
putation.
iHMYeKWMV
E73F6?21C@D<4
jZkMUHQeKWMV
fRYgTZhSeKWMV
GHIJKHLMNOGPMQHRSTIUQKLVSTLVHVS GWIXYMVUSQQZ[\TZUSLLZT
:
A?B<850C@D<4 7
/0121345678100 ;3<=16>5061>?@681> 9
]HLS^_‘STVMUHQ ]HLSa_bZTMcZYVHQ ]HLSN_dLQHYV
FIG. 6. Elementary cell of a 3D cluster state consist of 18
GUI]ZYYSUVMZY\HVVSTY
qubits. Universal resources for 3D MBQC is recursively-
generated by many elementary cells. The circles having an FIG. 7. (a)Schematic to perform 3D topological quantum
edge along the z-axis direction correspond to main qubits. computation in our scheme by using qubits embedded in a
Theothercirclescorrespondtosyndromequbitstodetecter- two-dimensional plane. (b) Unit cell to generate a bilayer
rors. The size of each layeron xy-planeis determinedby the 3D cluster state. We repeatedly put this cell in the two-
number of logical qubits and the depth of the error correc- dimensionalplane. Thelargecirclesdenotemainqubitswhile
tion codes. The height along thez-axis is determined by the small circles denote ancillary qubits. The edge between the
numberof quantumgates to beimplemented. qubitsdenotestheinteractionbetweenthem. (c)Cross-shape
structure composed of 4 main qubits and 5 ancillary qubits.
Wecanimplementacontrolled-phasegateoperationbetween
We can use either a cubic cluster state or bilayer clus-
an arbitrary pair of main qubitsin this structure.
ter state for the topological quantum computation, and
we choose the latter one to avoid an unnecesary deco-
herence effect, as previous authors did [18]. In this case,
A. Optimal interaction strength
it is necessary to generate the bilayer cluster state again
andagain,aftertheimplementationofthemeasurements
on one of the layers. We discuss how to generate a bi- Next, we estimate the optimal interaction strength of
layer 3D cluster state under the effect of always-onIsing theabovesystemforgeneratingathree-dimensionalclus-
interaction by using projective measurements and quan- ter state. We consider the same noise as in IIIE. For
tum feedforward. We again assume that the Ising type this estimation,weintroducethe followingsetup. Asde-
Hamiltonian described in Eq. 8 dominates this system, scribed in Fig. 7, we use three ancillary qubits inserted
and there is an interaction between every nearest neigh- betweenthetargetmainqubitsforperformingcontrolled-
bor qubit pairs. In our approach, unit cell to generate phase gate. We name these qubits as qubit A, B, C, D,
a bilayer three-dimensional cluster state is composed of and E where A and E denote the main qubits and B, C,
28 qubits (6 main qubits and 22 ancillary qubits), and and D denote ancillary qubits (See Fig. 8).
werepeatedlyputthesecellsonthesameplaneasshown Initially, all ancillary qubits are prepared in a ground
in the Fig. 7. Interestingly, although these qubits are lo- stateandmainqubitarepreparedinarbitrarystates. We
cated in a two-dimensional plane, it becomes possible to evaluate the error accumulation during the implementa-
implement a 3D topological quantum computation. tionof a controlled-phasegate in ourscheme using three
Letusconsiderqubitslocatedonthecross-shapestruc- ancillary qubits (Fig. 8). Firstly, by performing π pulse,
2
ture, which is a part of the unit cell (See the illustration we rotate the ancillary qubit C into + state, and sub-
| i
(c) in the Fig. 7 ). The 5 ancillary qubits are used to sequently rotate the other ancillary qubit B and D into
implement a controlled-phase gate between an arbitrary + state. In this case, since all nearest neighbor qubits
| i
pairoftwomainqubitsinthiscrossshapestructurewith- forthe qubitC arepreparedina groundstate,the qubit
out changing the states of the other main qubits. In- Cisnotaffectedbythecouplingstrengthfromanyother
terestingly, by preparing two of the ancillary qubits in qubitsandcanbeaccuratelyrotatedbyamicrowaveres-
groundstatesandpreparingtheotherancillaryqubitsin onant pulse. However, the qubit B (D) have unknown
the + stateasdescribedinFig.7,onlytwomainqubits energy shift due to the coupling from qubit A (E) and
| i
will be involvedin the implementation of the controlled- C during the rotation so that a detuning error occurs.
phase gate while the other main qubits do not affect the Secondly, we wait the appropriate time evolution of sys-
operationduetotheexistenceoftheancillaryqubitspre- tem with the Hamiltonian. During this time evolution,
paredinagroundstate,whichhasthesameanalogywith decoherence error of every qubit accumulates. Finally,
the case of a one-dimensional cluster state. we measure Yˆ and perform quantum feedforward on the
9
FIG. 8. Error accumulation during the implementation of a
controlled-phase gate between the main qubits. There are
twomainqubitsAandEinitiallypreparedinarbitrarystate.
Between these qubits, we insert three ancillary qubits B, C,
and D initially prepared in a ground state. As long as the
state of the nearest neighbor qubit contains a superposition,
we cannot determine a resonant frequency of the qubit due
to the always-on interaction, which induces a detuning error
ǫπ to rotate the qubit. Also, we assume a decoherence error FIG. 10. An achievable fidelity (F) of a controlled-phase op-
ǫ2that occurs during the time evolution to entangle nearest eration in ourschemeusingthreeancillary qubitsagainst in-
d
neighbor qubitsby theinteraction. teraction strength (gT2) and Rabi frequency (λT2).
8000 0.995 should be thousands times larger than that.
6000
VI. CONCLUSION
0.990
gT F
2
4000 Hereweshowascalablewaytogeneratetwoandthree-
dimensional cluster state with always-on Ising interac-
0.985
2000 tion. Here, we use projective measurements and quan-
tum feedforward to effectively turn on/off the interac-
tion in this system. Our schemes provide a novel way
150000 200000 250000 300000 to construct a surface code quantum computation and
λ topological quantum computation.
T
2
FIG. 9. The optimal coupling strength (gT ) and an achiev- Appendix A: Generating cluster state with
2
ablefidelityofacontrolled-phaseoperationinourschemeus- asymmetric interaction strength
ing three ancillary qubits against the Rabi frequency (λT ).
2
Thesolid line denotesF and thedashed line denotesgT . as
2 In this appendix, we discuss how to generate a cluster
with Fig. 4(a).
state with asymmetric coupling strength. Suppose that
only three qubits are involved in, and the other qubits
aresetnottointeractwiththesethreequbitsbycontrol-
qubitBandD,andsubsequentlyimplementthesameop-
ling the state of the ancillary qubits. As we described
eration on the qubit C. Again, due to the coupling from
before, by using unitary evolution, implementation of a
nearestneighborqubits,qubitBandDsuffersthedetun-
spinecho,quantummeasurementsandquantumfeedfor-
ing error for the feedforwardrotations while the qubit C
ward, we can perform a controlled-phase gate between
canbeaccuratelyrotatedbyaresonantmicrowavepulse.
twomainqubitswhereasingleancillaryqubitisinserted
To generate a whole three-dimensional cluster state, we
betweentwomainqubitsforthiscase. Thisisatwo-step
shouldperformthis typeofcontrolled-phasegates,verti-
elementaryoperationtoimplementcontrolled-phasegate
cally,horizontally,andaslantly,asshowninFig.7. Dur-
between main qubits under the effect of always-oninter-
ingthis3steps,additionaldecoherenceerroraccumulate.
action. We will use this operation recursively to make a
Therefore, we calculate and plot an achievable fidelity
large cluster state.
F =1 (4ǫπ +15ǫd) (44)
− 2
in Fig. 9. Also, we plot the optimal interaction strength 1. Generating one-dimensional cluster state
and an achievable fidelity against each λ in Fig. 4(b).
Since scalable topological quantum computation require First, we aim for generating a one-dimensional cluster
the error rate around below 1 %, this result shows that stateusingqubitsthatarearrangedinaraw. Inorderto
the Rabi frequency should be tens of thousands times avoid an exponentially large number of implementations
larger than the decay rate and the coupling strength ofπ pulses,weusethetwo-stepprocedureforgenerating
10
one-dimensional cluster state with m main qubits and
m 1 ancillary qubits as shown in Fig. 11. In this set-
−
ting,interactionstrengthsdifferfromeachother. Firstly,
po nml ut srq zy xwv (cid:127)~ }|{
FIG. 12. The procedure for generating a two-dimensional
clusterstateoverlogicalqubitsandsyndromequbits. Ateach
step,everymainqubitseffectivelyinteractswithatmost one
nearest-neighbor ancillary qubit, because the other nearest-
neighborancillaryqubitispreparedinagroundstatesothat
the interaction of these are effectively turned off. In this fig-
ure,largecirclesdenotemainqubitswhilesmallcirclesdenote
ancillary qubits. Main qubits are classified for three-typesof
qubits. Whitebigcircles denotelogical qubits,blue(red)big
FIG. 11. Due to the ancillary qubits prepared in a ground circles denotesyndromequbitsfor bit-flipdetection (dephas-
state, the number of qubits interacting with the main qubit ing detection).
is equal to or less than one so that we can use the scheme
described in Fig. 2. We create m bell pairs in the first step,
2
andcreateaone-dimensionalclusterstateinthesecondstep. state.
Thismakesthenecessarynumberofπ pulsesthesameasthe
numberof ancillary qubits.
3. Generating a three-dimensional cluster state
th
we performcontrolled-phaseoperationbetween2n 1
main qubit and 2nth(n = 1,2,..,m) main qubit so−that In this subsection, we suggest a procedure to make a
2
m bell pairs are created. Since the 2kth(k = 1,2,..,m) three-dimensional cluster state as shown in Fig. 13. For
2 2
ancillary qubits are prepared in a ground state, we only
need to consider the dynamics of three-qubit during the
time evolution, and so we can apply the scheme to turn
on/off effective interaction for three-qubit case as de-
scribed in Fig. 1. Secondly, we perform controlled-phase
th th
operations between 2n main qubit and 2n+1 main
qubit so that a one-dimensionalcluster state can be cre-
th
ated. Here, the 2k 1 ancillaryqubits are ina ground
th − FIG. 13. Only main qubits are presented in this figure.
state, and 2k ancillary qubits interact with two main
Firstly, we generate one-dimensional cluster states using log-
qubits. Again,onlythreequbitsinteractwitheachother,
ical and syndrome qubits on the same column. Secondly, we
and so we can apply the scheme described in Fig. 1. generatetwoseparabletwo-dimensionalgraphstates. Finally,
we generate a three-dimensional cluster state.
2. Generating a two-dimensional cluster state this procedure,weuse 3m2 mainqubits and 11m2 ancil-
2 2
laryqubitsthatarearrangedonatwo-dimensionalplane
Next, we aim for generating a two-dimensional clus- with asymmetric coupling strength. Here, we show how
ter state using m2 main qubits and 2m(m 1) ancillary toperformcontrolled-phaseoperationbetweentwo-main
−
qubits thatarearrangedontwo-dimensionallattice with qubits A and E via three-ancillary qubits B, C, and
asymmetric coupling strength. For this purpose,we sug- D with asymmetric interactions as in Fig. 14. Firstly,
gestafour-stepprocedureasshowninFig.12. Firstly,we we perform controlled-phase operation between qubit A
generate m Bell-pairsbetweeneachnearestpairsof logi- and C using our technique in Fig. 2. Secondly, we per-
2
cal and syndrome qubits at each column. At every step, form controlled-phase operation between qubit C and
ancillary qubits, which are not used for controlled-phase E using the same technique. Finally, we perform Yˆ
operations, prepared in ground state. Secondly, we per- basis measurement quantum feedforward operations on
formcontrolled-phaseoperationsbetweeneachBell-pairs qubitC sothatcontrolled-phaseoperationcanbeimple-
in the same column to generate one-dimensional cluster mented between qubit A and E. We use this three-step
states. Thirdly, we perform controlled-phase operations controlled-phase operation recursively to make a large
in horizontal direction. At each controlled-phase opera- three-dimensional cluster state in following procedure.
tion, the syndrome qubit is arrangedat the left side and The details of procedure for generating a three-
the logical qubit is arranged at the right side. Finally, dimensionalclusterstatearedescribedasfollows. Firstly,
we perform controlled-phase operation between all the wegeneratem-qubitone-dimensionalclusterstatescom-
remaining nearest-neighborpair of logicaland syndrome posedof m logicalqubitsand m syndromequbitsinver-
2 2
qubits so that we can obtain a two-dimensional cluster tical direction. For these operations, we use the same