Bounds on quantum process fidelity from minimum required number of quantum state fidelity measurements

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by  Jaromir Fiurasek

Bounds on quantum process fidelity from minimum required number
of quantum state fidelity measurements
Jarom´ır Fiura´ˇsek and Michal Sedla´k
Department of Optics, Palack´y University, 17. listopadu 1192/12, CZ-771 46 Olomouc, Czech Republic
To certify that an experimentally implemented quantum transformation is a certain unitary op-
eration U on a d-dimensional Hilbert space, it suffices to determine fidelities of output states for
d+1 suitably chosen pure input states [Reich et al., Phys. Rev. A 88, 042309 (2013)]. The set
of these d+1 probe states can consist of d orthogonal states that form a basis and one additional
state which is a balanced superposition of all d basis states. Here we provide an analytical lower
bound on quantum process fidelity for two-qubit quantum gates which results from the knowledge
of average statefidelity for thebasis states and thefidelity of thesuperposition state. Wecompare
4
this bound with the Hofmann bound that is based on knowledge of average state fidelities for two
1
mutually unbiased bases. We also discuss possible extension of our findingsto N-qubit operations.
0
2
PACSnumbers: 03.65.Wj,03.67.-a

n
a
J I. INTRODUCTION number of d+1 pure probe states turns out to be more

3 difficultduetoreducedsymmetry. InRef. [10],thisprob-
2 lem was studied numerically and althoughan expression
Development and testing of advanced quantum infor-
quantifying the quantum gate performanceas a function
] mation processing devices requires efficient methods for
h of state fidelities was proposed, it was noted that it can
theircharacterization. Fullquantumprocesstomography
p lead to underestimation of gate error in certain cases.
[1–3] is suitable for small-scaledevices such as two-qubit
- Herewe deriveanexactlowerboundonquantumpro-
t quantum gates. However, the number of measurements
n cess fidelity of two-qubit operations based on knowledge
a that need to be performed grows exponentially with the of the average state fidelity F for certain basis j and
u numberofqubitswhichmakesthisapproachrathertime- | i
state fidelity G for a state s which is a balanced su-
q consuming and impractical for larger systems. There- | i
perposition of all the basis states j . We compare this
[ fore, increasing attention has been paid in recent years boundwiththeHofmannboundwh|oisedeterminationre-
1 to development of alternative less demanding techniques quires measurement of state fidelities for two bases and
v for assessment of the performance of quantum devices
wefindthatthenewboundistypicallymuchweakerthan
4 [4–10]. Typically, the device is probed by various input
the Hofmann bound. Therefore, the number of probe
6 states, and measurements are performed on the output
9 states can be reduced from 2d to d+1 only at a cost of
states. In this context, one may ask what is the min-
5 potentially much less precise device characterization.
imum necessary number of input states to certify that
. The rest of the paper is organized as follows. In Sec-
1 the implemented quantum operationis a certain unitary
0 operation U on a d-dimensional Hilbert space. Very re- tionIIweprovideanexplicitconstructionofatwo-qubit
4 cently, Reich et al. showed that d+1 pure input probe quantum operation (a trace-preserving completely posi-
1 tivemap)whichforgivenstatefidelitiesF andGachieves
states are sufficient for this purpose [10]. A suitable set
: minimum quantum process fidelity. This fidelity thus
v ofthesestatesconsistsofdbasisstatestogetherwithone
Xi additional state which is a superposition of all the basis provides a lower bound on process fidelity of any opera-
tion achieving state fidelities F and G. Analytical proof
states. If the output state fidelities for these d+1 input
r of this bound is provided in Section III and extension
a states are all equal to 1, then the implemented quantum
of our construction to N-qubit operations is proposed
operation must be exactly the target unitary operation
in Section IV. Although we do not provide any rigorous
U [10].
optimality proof for the N-qubit case, our construction
An appealing feature of this approach is that it re- nevertheless illustrates that the gap between the fidelity
quires only d+1 input probe states in comparison to d2 bound and the true fidelity will typically increase fast
probe states necessary for full quantum process tomog- with the growing number of qubits. Finally, Section V
raphy. These results are similar in spirit to the earlier contains a brief summary and conclusions.
findingsbyHofmann[4],whoderivedananalyticallower
bound on quantum process fidelity F in terms of aver-
χ
age state fidelities F and F for two mutually unbiased
′ II. TWO-QUBIT OPERATIONS
bases, F F +F 1. Clearly, if F =F =1 then also
χ ′ ′
≥ −
F =1. TheHofmannboundwassuccessfullyappliedto
χ
characterization of several experimentally implemented Let 0 and 1 denote the computational basis states
| i | i
two-qubit and three-qubit quantum operations [11–17]. of a single qubit and define superposition states =
Finding a similar analytical lower bound on quantum 1 (0 1 ). We wouldlike to determine a lowerb|±ouind
√2 | i±| i
process fidelity also for the scheme with the minimum onfidelity ofa two-qubitquantumoperation withuni-
E
2
tary operation U provided that we know the average Similarly, fidelity of the output state ρ+ obtained from
out
output state fidelity F for the computational basis jk , input state ++ is defined as
| i | i
where j,k 0,1 , and also output state fidelity G for
input state∈+{+ w}hichis abalancedsuperpositionofall G=h++|U†ρ+outU|++i. (9)
| i
computational basis states,
We can express this fidelity as
1 G=Tr[(I U)R (I U )χ], (10)
++ = (00 + 01 + 10 + 11 ). (1) G †
⊗ ⊗
| i 2 | i | i | i | i
where
According to the Choi-Jamiolkowski isomorphism [18,
19], any quantum operation can be represented by a RG = ++ ++ ++ ++ . (11)
E | ih |⊗| ih |
positivesemidefinite operatorχonthe tensorproductof
Ourgoalis todeterminealowerboundonquantumpro-
input and output Hilbert spaces. Given input state ρin, cess fidelity
the output state can be calculated according to
Tr[χ χ]
U
ρout =Trin[ρTin⊗Ioutχ], (2) Fχ = Tr[χU]Tr[χ] (12)
where Trin denotes the partial trace over the input from the knowledge of state fidelities F and G. For a
Hilbert space, I denotes the identity operator, and T two-qubit unitary operation U we explicitly have
stands for transposition in the computational basis.
1
We shall consider deterministic operations described by F = χ χχ . (13)
χ U U
4h | | i
trace-preserving maps. The trace-preservationcondition
can be expressed as Weshallproceedbyconstructingaparticularquantum
operation χ˜ that achieves the state fidelities F and G.
Trout[χ]=Iin, (3) We then prove that the quantum process fidelity of this
particular operation provides a lower bound on F , i.e.
χ
anditguaranteesthatTr[ρout]=Tr[ρin]forarbitraryρin. it represents the lowest possible value of F consistent
χ
Inthisformalism,aunitaryoperationU isisomorphicto with F and G. Our ansatz for the quantum operation χ˜
a pure maximally entangled state,
reads
χU =4|χUihχU|, (4) χ˜=(I⊗U)χ˜S(I⊗U†), (14)
where χU =(Iin U) Φ+2 , where
| i ⊗ | i
3
1
1 χ˜ = χ χ (15)
Φ+ = jk jk (5) S | mih m|
| 2i 2 | i⊗| i mX=0
jX,k=0
and
is a maximally entangled state between qubits in the in-
put Hilbert space andoutput Hilbert space, andthe fac- χ0 = aZ00 Φ+2 +b ++ ++ ,
| i | i | i| i
tor 4 ensures correct normalization of χ as implied by
the trace-preservationcondition (3). U |χ1i = cZ01|Φ+2i+d|++i|+−i,
Theaverageoutputstatefidelityforcomputationalba- χ2 = cZ10 Φ+2 +d ++ + ,
sis is defined as | i | i | i|− i
1 |χ3i = cZ11|Φ+2i+d|++i|−−i.
1
F = 4jX,k=0hjk|U†ρjokutU|jki, (6) Hfineerde aZsjkσ==Iin0⊗0σZj ⊗1σZk1,,tσh0e=PaIu,lainmdaσtr1ix=σσZ .isTdhee-
Z | ih |−| ih | Z Z Z
complete positivity condition χ˜ 0 is satisfied by con-
where U jk is a pure output state that would be gener- ≥
| i struction. On inserting the ansatz (14) into Eq. (3), we
atedbytheunitaryU andρjokut =Trin[jk jk Ioutχ]is find after some algebra that the trace-preservation con-
| ih |⊗
the output state produced by the actually implemented
dition is equivalent to the following constraints:
quatum operation χ. On inserting the formula for ρjk
out
into Eq. (6), we obtain [17] a2+3c2 = 4,
b2+ab+3d2+3cd = 0. (16)
F =Tr[(I U)RF(I U†)χ], (7)
⊗ ⊗
The output state fidelities F and G can be determined
where by inserting χ˜ into Eqs. (7) and (10), respectively. We
get
1
1
RF = 4 |jkihjk|⊗|jkihjk|. (8) F = 1 (2a+b)2+ 3 (2c+d)2, (17)
jX,k=0 16 16
3
and
(2b+a)2
G= . (18)
4
Finally, with the help of Eq. (13) we can also calculate
the quantumprocessfidelity ofoperationχ˜ with unitary
operation U,
(2a+b)2
F˜ = . (19)
χ
16
The system of equations (16), (17), and (18) can be
solvedandtheparametersa,b,c,anddcanbeexpressed
in terms of the output state fidelities,
a = 2 (8F 5)√G 4 (1 F)(4F 1)(1 G) , FIG. 2: Lower bound on quantum process fidelity Fχ deter-
3h − − p − − − i mined from the knowledge of state fidelities F and G (solid
a
b = √G , line) and the Hofmann lower bound on quantum process fi-
− 2 delity (dashed line) are plotted for two-qubit quantum oper-
4 a2 ations assuming G=F′ =F.
c = − ,
r 3
1 G 1 4 a2
d = r −3 − 2r −3 . (20) ThefidelitythresholdFth isdeterminedbythecondition
that F˜χ = 0 when F = Fth. If F < Fth then the state
Oninsertingthe expressionsforaandbintoEq. (19)we fidelities F and G are consistent with F =0. In Fig. 1
χ
finally arrive at a formula for F˜ as a function of F and weplotthe lowerboundonF asafunctionofF andG.
χ χ
G, We emphasize that our calculation is valid for arbitrary
two-qubitunitaryoperationU. Therefore,the boundF˜
2 χ
F˜ = (2F 1)√G (4F 1)(1 F)√1 G . is also universally valid.
χ
h − −p − − − i(21) It is instructive to compare this lower bound with the
We prove in the next Section that F˜ provides a lower Hofmann bound Fχ F +F′ 1 which is based on the
χ ≥ −
knowledgeofaverageoutput state fidelities F andF for
bound on the quantum process fidelity F providedthat ′
χ
two mutually unbiased bases. To make such comparison
F Fth, (22) possible, we shall assume that the average output state
≥
fidelity F for basis which is mutually unbiased with the
′
where computational basis and which contains state ++ is
| i
1 equal to the fidelity G. Let us consider high-fidelity op-
Fth = 5 G+ 9 10G+G2 . (23) eration, F = 1 ǫ and G= 1 δ, where ǫ,δ 1. If we
8(cid:16) − p − (cid:17) − − ≪
keep only terms up to linear in δ and ǫ, we obtain
F 1 ǫ δ (24)
χ
≥ − −
1
for the original Hofmann bound, and
0.8
0.6
Fχ 0.4 Fχ &1−4ǫ−δ−2√3√ǫδ (25)
0.2
forthe bound(21). We canseethattheHofmannbound
0 ishigherthanthe bound(21)andthegapincreaseswith
1
0.8 1 decreasing fidelity F. This is illustrated in Fig. 2 where
0.6 0.9
0.4 0.8 we plot both bounds as a function of F assuming that
G 0.2 0.6 0.7 F F = G. To provide some scale we note that, for in-
0 0.5
stance, fidelity of two-qubit cnot gate with the identity
operation reads 0.25. We can thus conclude that the
FIG. 1: Lower bound on quantum process fidelity Fχ of a state fidelities F and G have to be very high to obtain a
two-qubitquantumoperationisplottedasafunctionofstate meaningfulandnontrivialboundonprocessfidelityfrom
fidelities F and G. Eq. (21).
4
III. OPTIMALITY PROOF Ifwe solvethis systemofequationsandexpressb, c,and
d as functions of a and G, c.f. Eq. (20), we obtain
Here we prove that the two-qubit quantum operation
√4 a2(3a+2√G)
(14) constructed in the previous section exhibits mini- x = − ,
mumquantumprocessfidelityF compatiblewithF and 2a√1 G 2 G(4 a2)
χ − − −
G. The proof is based on the techniques from semidefi- p 3a+2√G
nite programming [20, 21]. We define an operator w = 3 4 a2+2√1 G ,
−(cid:16) p − − (cid:17)64 G(1 G)
−
M = 14|Φ+2ihΦ+2|+xRF +wRG+yI+z|++ih++|⊗Iout. y = 312(cid:16)3a+2√G(cid:17) 3G√(44−aa22)+p2√a√11−GG,
(26) − − −
The parameters appearing in definition of M can be in- √4 a2 2√1 pG
z = − − − y.
terpretedasLagrangemultipliersthataccountforafixed 2√1 G
valueofF andG,andforthetracepreservationcondition −
SincetheconditionMχ˜ =0issatisfiedbyconstruction,
(3). Suppose that we choose the Lagrange multipliers S
it remains to prove that M 0. The eigenvalues of
such that ≥
operator M are listed below,
(I⊗U)M(I⊗U†)χ˜=0, (27) λ1 =y,
1 1
and λ2 = (A √B), λ3 = (A+√B), (34)
8 − 8
1 1
M ≥0. (28) λ4 = 8(C−√D), λ5 = 8(C+√D).
Note that the condition (27) can be equivalently ex- Here
pressed as
A=x+8y+4z, B =x2 4xz+16z2, (35)
−
Mχ˜ =0, (29)
S
C =1+4w+x+8y+4z, (36)
hence it is independent on the unitary operation U. It
follows from the inequality M 0 that and
≥
Tr (I U)M(I U†)χ 0, (30) D =1+16w2+2x+x2−4w(1+x−8z)−4z−4xz+16z2.
⊗ ⊗ ≥ (37)
(cid:2) (cid:3)
for arbitrary trace preserving map χ, because trace of The eigenvalue λ1 is 8-fold degenerate and the eigenval-
productoftwopositivesemidefinite operatorsisnonneg- uesλ2 andλ3 areeach3-folddegenerate. One canverify
by direct calculation that A2 = B and C2 = D. The
ative. On inserting the expression (26) into Eq. (30) we
operator M is thus positive semidefinite if y 0, A 0,
obtain ≥ ≥
and C 0. After some algebra we find that
≥
Fχ+xF +wG+4y+z ≥0. (31) 3a+2√G 16 3a2 4G
A= − − , (38)
By taking trace of Eq. (29) we find that F˜χ = [xF + 16√1−G G(4−a2)−a√1−G
− p
wG+4y +z]. Therefore, the positive semidefiniteness and
of M together with the condition (29) implies a lower
bound on quantum process fidelity, 3a2+4G 3√4 a2+2√1 G
C = − − . (39)
16√G G(4 a2) a√1 G
Fχ F˜χ. (32) p − − −
≥ We shall first derive several useful auxiliary inequali-
In what follows we provide explicit formulas for the La- ties. Since F 1, it follows from Eq. (20) that
≤
grange multipliers and prove that M 0.
≥ a 2√G. (40)
The Lagrange multipliers can be determined from Eq. ≤
(29)whichisequivalentto4conditionsM χm =0,m= AssumingafixedG,aisamonotonicallyincreasingfunc-
| i
0,1,2,3. This providesa systemof4 linearequationsfor tion of F in the interval F [Fth,1]. Parameter a
the 4 unknown parameters x, w, y, and z, as a function of F exhibits a∈single local minimum at
F0 =(5 3√G)/8 and F0 Fth for all G [0,1]. Mini-
2a(1+x+4y)+b(1+x) = 0, mum val−ue of a for a fixed≤G in the interva∈l F [Fth,1]
∈
a(w+z)+2b(w+y+z) = 0, is thus achieved at F =Fth and we have
2c(x+4y)+dx = 0, (33)
2√G
cz+2d(y+z) = 0. a . (41)
≥− 3
5
Recall that we have to restrict ourrselves to F Fth Similarly, we can define fidelity G of output state ob-
≥
because for F <Fth the operation(14) does not provide tained from the input superposition state
the minimum quantum process fidelity compatible with
given F and G. Inequalities (40) and (41) imply that 2N 1
1 −
a2 4G, hence s = j , (45)
≤ | i √2N | i
Xj=0
3a2+4G 16G 16. (42)
≤ ≤ and we have G=Tr[s s U s sU χ].
†
| ih |⊗ | ih |
Furthermore,itfollowsfromtheinequalitya 2√Gthat Analogicallyto the two-qubitcase we constructanN-
≤ qubit quantum operation,
G(4−a2)−a√1−G≥0. (43) χ˜=(I U)χ˜S(I U†), (46)
p ⊗ ⊗
The inequalities y 0, A 0 and C 0 now directly where
≥ ≥ ≥
follow from the inequalities (41), (42) and (43). This
proves that M ≥0 in the entire domain F ≥Fth. 2N−1
For the sake of completeness, we also explicitly show χ˜S = χj χj . (47)
| ih |
thatifF <Fththenthelowerboundonquantumprocess Xj=0
fidelity reads F˜ = 0. It is sufficient to find quantum
χ
The constituents χ can be expressed as superposi-
operations with F =0 for three boundary points F =0 j
χ | i
tions of maximally entangled states and product states
and G = 1, F = 1 and G = 0, and F = 0 and G = 0.
of qubits in intput and output Hilbert spaces. For j =0
At any point in the area F < Fth a quantum operation
with given F, G and F˜ = 0 can then be constructed as we have
a mixture of these three operations and operations (14)
correspondingtotheboundarylineF =Fth,whereF˜χ = |χ0i=a|Φ+Ni+b|si|si, (48)
0. Fidelities F = 0, G = 1, and F = 0 can be achieved
χ while for j 1 we define
byaunitaryoperationUX,whereX =σ σ performs ≥
X X
⊗
a bit flip on each qubit, σX = 0 1 + 1 0. Similarly, χ =I V c Φ+ +d s s . (49)
F =1, G=0, and F =0 is ac|hiiehve|d b|yiohp|eration UZ, | ji ⊗ j | Ni | i| i
χ (cid:0) (cid:1)
where phase flips are inserted before U, Z = σZ σZ. Here
⊗
Finally, if we combine both bit flips and phase flips we
obtain unitary operation UZX which exhibits F = 0, 2N 1
G=0, and F =0. Φ+ = 1 − j j (50)
χ | Ni √2N Xj=0 | i| i
is a maximally entangledstate of2N qubits, and the N-
IV. N-QUBIT OPERATIONS
qubit unitary operatorsV are defined as products of σ
j Z
and identity operators,
In this section we generalize the constructionof quan-
tum operation (14) to N-qubit operations. Note that in N
V = σjk. (51)
contrasttothetwo-qubitoperationswedonotprovethat j Z
the resulting expressionfor quantum processfidelity is a Ok=1
lowerboundonF . Nevertheless,thisapproachallowsus
to investigate theχscaling of the bound with the number The trace preservation condition Trout[χ˜] = Iin is
equivalent to
ofqubits. Inparticular,weshallshowthatunderreason-
able assumptions the gap between the Hofmann bound a2+(2N 1)c2 =2N,
and the bound determined by average state fidelity F −
b2+21 N/2ab+(2N 1) d2+21 N/2cd =0, (52)
andfidelityofasinglesuperpositionstateGgrowsexpo- − −
−
nentially with the number of qubits N. (cid:0) (cid:1)
and the fidelities F and G can be expressed as
We shall label the N-qubit computational basis states
by an integer j, 0 j 2N 1. We define j = 2 2
≤ ≤ − | i 1 b 1 d
|j1i|j2i···|jNi, wherejk denoteskthdigitofbinaryrep- F = 2N (cid:18)a+ 2N/2(cid:19) +(cid:18)1− 2N(cid:19)(cid:18)c+ 2N/2(cid:19) ,
resentation of integer j. With this notation at hand, we
can define the average state fidelity F in the computa- a 2
G = +b ,
tional basis, (cid:16)2N/2 (cid:17)
(53)
2N 1
1 −
F = Tr j j U j j U†χ . (44) Recall that N-qubit unitary operation U is represented
2N Xj=0 (cid:2)| ih |⊗ | ih | (cid:3) by χU = 2N|χUihχU|, where |χUi = I⊗U|Φ+Ni. If we
6
insertthe N-qubitoperationsχ andχ˜ into the formula V. CONCLUSIONS
U
for quantum process fidelity, Eq. (12), we obtain
2 Insummary,wehavederivedalowerboundonfidelity
1 b
F˜χ = 2N (cid:18)a+ 2N/2(cid:19) . (54) of two-qubit quantum gates imposed by the knowledge
ofaveragestatefidelityF foronebasisandknowledgeof
statefidelityGforoneadditionalbalancedsuperposition
Note, that for N =2 we recover the formulas derived in
state. In our calculations we have explicitly considered
SectionII.Withthehelpoftheexpressions(52)and(53)
the computational basis but in practice this basis may
we obtain after some algebra formula for the quantum
process fidelity F˜ as a function of the state fidelities F be arbitrary, because any basis transformation can be
χ
included into the unitary operation U. We have seen
and G,
that the quantum gate characterization with the mini-
F˜χ = 1 (1 F)2N−1 √G mum number of pure probe states would generally yield
n(cid:2) − − (cid:3) rather low bound on process fidelity. This bound is par-
2 ticularly sensitive to the value of the average state fi-
− (1−F)(1−G)q2N −1−(1−F)22N−2(cid:27) . delity F. In any potential experimental application of
p this technique one should therefore choose the basis for
(55)
which one expects the best performance. At a cost of
doubling the number of probe states one could instead
The quantum process fidelity F˜χ vanishes for F = Fth, determinethe originalHofmannboundthatwillbetypi-
where
cally significantly higher. Therefore, characterizationby
1 1 G 2 theminimumnumberofprobestateswouldbesuitablein
Fth =1−2N−1+22−N−1+22Nq(1−G)[(2N −1)2−G]. situationswherethegateexhibitshighfidelityandwhere
(56) measurements for the basis states are easy to implement
The lower bound on quantum process fidelity will thus whilemeasurementsforthesuperpositionstatesarevery
certainly be zero when F Fth. It follows that with difficult and demanding.
≤
increasing number of qubits N the fidelity F has to be
exponentiallycloseto1,F 1 21 N,inordertoobtain
−
≥ −
a nontrivial lower bound on F . Assuming fixed state
χ
fidelities F andG,the boundonF determinedbythese
χ
fidelities will thus quickly become very low with growing
Acknowledgments
N. This should be contrasted with the Hofmann bound
F F +F 1 whose form does not depend on the
χ ′
≥ −
numberofqubitsN. WestressthatforN >2wedidnot J.F.acknowledgessupportbytheCzechScienceFoun-
provethatF˜χistheultimatelowerboundonFχforgiven dation (Project No. 13-20319S). M.S. acknowledges
F andG. However,anysuchultimateboundcanonlybe support by the Operational Program Education for
smaller than F˜χ. Therefore, our conclusions concerning Competitiveness - European Social Fund (project No.
the scaling with N would hold even if F˜ would not be CZ.1.07/2.3.00/30.0004) of the Ministry of Education,
χ
the ultimate lower bound on F . Youth and Sports of the Czech Republic.
χ
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