Table Of ContentVerified measurement-based quantum computing with hypergraph states
Tomoyuki Morimae,1,∗ Yuki Takeuchi,2,† and Masahito Hayashi3,4,‡
1ASRLD Unit, Gunma University, 1-5-1 Tenjincho Kiryushi Gunma, 376-0052, Japan
2Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
3Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan
4Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117542, Singapore
Hypergraph states are generalizations of graph states where controlled-Z gates on edges are re-
placed with generalized controlled-Z gates on hyperedges. Hypergraph states have several advan-
tagesovergraph states. Forexample, certain hypergraphstates, suchastheUnionJack states, are
universalresourcestatesformeasurement-basedquantumcomputingwithonlyPaulimeasurements,
7 while graph state measurement-based quantum computing needs non-Clifford basis measurements.
1 Furthermore, it is impossible to classically efficiently sample measurement results on hypergraph
0 states with a constant L1-norm error unless the polynomial hierarchy collapses to the third level.
2 Although several protocols have been proposed to verify graph states with only sequential single-
qubit Pauli measurements, there was no verification method for hypergraph states. In this paper,
n
we propose a method for verifying hypergraph states with only sequential single-qubit Pauli mea-
a
surements. Asapplications, weconsiderverifiedblindquantumcomputingwith hypergraphstates,
J
and quantumsupremacy demonstrations with hypergraph states.
0
2
Many-point correlations in quantum many-body sys- quantumcomputingtoaremoteserver[27,28]. Thehon-
]
h tems are one of the most essential ingredients in est server sends each qubit of a graph state one by one
p condensed-matter physics and statistical physics. Corre- totheuser,andusercanrealizeanyquantumcomputing
-
t lationsofsequentialsingle-qubitmeasurementsonquan- with only sequential single-qubit measurements. If the
n
tumstatesarealsoimportantdriveforcesforquantumin- server is malicious, however, a completely wrong state
a
formation processing. For example, measurement-based mightbesenttotheuser. Theuserthereforeneedstotest
u
q quantum computing [1], which is nowadays one of the the state sent from the server. In such a quantum cryp-
[ standard quantum computing models, enables univer- tographicscenario,thesituationisworsethanthesingle-
sal quantum computing with only adaptive single-qubit party laboratory experiments, since the noises on the
1
v measurements on certain quantum states, such as graph givenstate arecausedby maliciousserversandtherefore
8 states [1] and other condensed-matter-physically moti- not necessarily physically natural ones. Severalmethods
8 vated states including the AKLT state [2–17]. Further- ofverifyinggraphstateswithonlysequentialsingle-qubit
6
more, not only adaptive but also non-adaptive single- Pauli measurements have been proposed [28, 29]. (If
5
qubit measurements on graph states can demonstrate a more than two non-communicating servers are available,
0
. quantumnesswhichcannotbeclassicallyefficientlysimu- acompletelyclassicalusercanverifystabilizerstates[30–
1
lated: itisknownthatifprobabilitydistributionsofnon- 32].) In the protocol of Ref. [28], the user does a test so
0
adaptive sequential single-qubit measurements on graph called the stabilizer test on some parts of the state sent
7
1 states are classically efficiently sampled, then the poly- fromtheserver. Thestabilizertestcanbedonewithonly
: nomial hierarchy collapses to the third level [18–20] or sequential single-qubit Pauli measurements. If the user
v
i the secondlevel[21]. Thepolynomialhierarchyisahier- passes the test, the remaining state is guaranteed to be
X
archy of complexity classes generalizing P and NP, and close to the ideal graph state.
r it is not believed to collapse in computer science. It is
a
an example of recently well studied “quantum suprema- Since the protocol of Ref. [28] makes no assumption
cies”ofsub-universalquantumcomputingmodels,which (such as the i.i.d. sample or physically natural noises)
are expected to be easier to experimentally implement, on the given state, the verification method can be used
butcanoutperformclassicalcomputing. (Fordetails,see in quantum cryptographic contexts. In particular, ver-
Refs. [18–24] and their supplementary materials.) ified blind quantum computing and verified quantum
supremacy demonstrations can be realized with graph
For practical implementations of measurement-based states verified through the protocol. There are, how-
quantumcomputingandexperimentaldemonstrationsof ever, two problems. First, in the verified blind protocol
the quantum supremacy, verifying graph states is essen- of Ref. [28], the user needs non-Clifford basis measure-
tial,sinceinrealityageneratedstatecannotbethe ideal ments for computing (the verification itself can be done
graph state due to some experimental noises. The prob- with only Pauli measurements). It would be better if
lembecomesmoreseriousifweconsiderdelegatedsecure both the verification and the computation can be done
quantum computing, so called blind quantum comput- withonlyPaulimeasurements[33]. Second,thequantum
ing [25, 26]. It is known that the ability of sequentially supremacy demonstration with graph states [18], which
measuring single qubits is enough to secretly delegate needs only non-adaptive measurements, requires some-
2
howastrictapproximation,namelyamultiplicative-error The stabilizer g of |Gi associated with the vertex i is
i
approximation. defined by
Recently, two breakthroughs that solve these draw-
backs of graph states have been done. These results g ≡ CZ X CZ
i e i e
use hypergraph states [34–38] in stead of graph states. (cid:16)eY∈E (cid:17) (cid:16)eY∈E (cid:17)
g g
(For the definition of hypergraph states and their prop-
= X Z CZ ,
erties, see below.) First, certain hypergraph states, such i j j,k
astheUnionJackstates,areuniversalresourcestatesfor (cid:16)j∈YWiZ (cid:17)(cid:16)(j,k)Y∈WiCZ (cid:17)
measurement-basedquantumcomputing withonlyPauli
where
measurements [39]. This result solves the first problem,
namely, the requirement of non-Clifford basis measure- WZ ≡ {j ∈V | (i,j)∈E},
i
ments for the user. Therefore, by using the hypergraph
WCZ ≡ {(j,k)∈V ×V | (i,j,k)∈E}.
states,theone-waysecuredelegatedquantumcomputing i
is possible for the user who can do only Pauli measure-
It is easy to check that the following properties are sat-
ments. Ref. [39] also pointed out that hypergraphstates
isfied: [g ,g ] = 0 for all i,j ∈ V. g |Gi = |Gi for all
i j i
are important in the study of symmetry-protected topo- i∈V. g2 =I⊗n for all i∈V. n I⊗n+gi =|GihG|.
logical orders. Second, it was shown in Ref. [19] that if i i=1 2
hypergraphstatesareconsidered,themultiplicativeerror Stabilizer test for gi.— BefoQre introducing our verifi-
cation protocol, we define the stabilizer test for each g ,
requirementcanbereplacedwithanL1-normone,which i
which is an essential ingredient of the protocol. Note
is more relaxed. This result solves the second problem.
that CZ = 1(I ⊗I +I ⊗Z +Z ⊗I −Z ⊗Z ).
In short, hypergraph states are promising novel re- j,k 2 j k j k j k j k
Therefore
source states for many quantum information processing
tasks. Unfortunately, however, there was no verification 1
g = X Z σ (t )
method for hypergraph states. In particular, we did not i i j 2r j,k j,k
know how to test a given hypergraphstate with only se- (cid:16)j∈YWiZ (cid:17)(cid:16) t∈{1X,2,3,4}r(j,k)Y∈WiCZ (cid:17)
quential single-qubit Pauli measurements. It was a huge 1
= s ,
obstacle for practical applications of hypergraph states 2r t
t∈{1X,2,3,4}r
in quantum information and condensed matter physics.
peIrngrtahpihs pstaapteers,wwiethproonplyoseseaqumenettihaoldsifnogrlev-eqruifbyiitngPahuyl-i where r ≡|WiCZ|, t≡{tj,k}(j,k)∈WiCZ, σj,k(1)≡Ij ⊗Ik,
σ (2)≡I ⊗Z , σ (3)≡Z ⊗I , σ (4)≡−Z ⊗Z ,
j,k j k j,k j k j,k j k
measurements. As in the case of the graph state veri-
and
fication [28], the user does a certain test on some parts
of the state sent from the server. If the user passes the s ≡ X Z σ (t ) .
t i j j,k j,k
test, then the remaining state is guaranteed to be close (cid:16)j∈YWiZ (cid:17)(cid:16)(j,k)Y∈WiCZ (cid:17)
to the ideal hypergraph state. As applications, we con-
siderverifiedblindquantumcomputingwithhypergraph Let us define a bit α ∈{0,1} and a subset D ⊆V such
t t
states, and verified quantum supremacy demonstrations that
with hypergraph states.
Hypergraphstates.—Wefirstdefinehypergraphstates, s =(−1)αtX Z .
t i j
and explain their properties. A hypergraph G ≡ (V,E) (cid:16)jY∈Dt (cid:17)
is a pairofa setV ofverticesandasetE ofhyperedges,
Note that α and D can be calculated in polynomial
where n ≡ |V|. A hyperedge may link more than two t t
time. In fact, α can be calculated in the following way.
vertices. For simplicity, in this paper, we assume that t
We first set α = 0. If t = 4, we flip α . We do it
2 ≤ |e| ≤ 3 for all e ∈ E, where |e| is the number of t j,k t
for all (j,k) ∈ WCZ. Since |WCZ| ≤ n−1 = O(n2), it
vertices linked to the hyperedge e. (Generalizations to i i 2
other cases would be possible.) Let takes at most polynomial time. Furthe(cid:0)rmo(cid:1)re, Dt can be
calculated in the following way. We first set D = WZ.
t i
|Gi≡ CZe |+i⊗n We then update Dt according to tj,k for each (j,k) ∈
(cid:16)eY∈E (cid:17) WiCZ. Again, |WiCZ| ≤ O(n2) means that it takes at
g
most polynomial time.
be the hypergraph state corresponding to the hyper-
Let ρ be an n-qubit state. We define the “stabilizer
graph G, where CZ ≡ I −2 |1ih1| is the
e i∈e i i∈e i test for g on ρ” as the following Alice’s action:
generalized CZ gate actNing on vertiNces in the hyper- i
g
edge e. Here, I is the two-dimensional identity oper- 1. Alice randomly generates t∈{1,2,3,4}r.
ator. For example, if |e| = 2, it is nothing but the
standard CZ gate. If |e| = 3, it is the CCZ gate, 2. She measures ith vertex of ρ in X, and jth vertex
CCZ ≡(I⊗2−|11ih11|)⊗I +|11ih11|⊗Z. of ρ in Z for all j ∈D .
t
3
Let x ∈ {+1,−1} be the measurement result of the X 2. Soundness: if Alice accepts Bob, the state ρ
comp
measurement, and z ∈ {+1,−1} be that of the Z mea- of the computing register satisfies hG|ρ |Gi ≥ 1− 1
j comp n
surementonvertexj ∈D . We saythatAlice passesthe with a probability larger than 1− 1.
t n
stabilizer test for g on ρ if x z =(−1)αt. Proof of the completeness.— We first show the com-
i j∈Dt j
The probability ptest,i thatQAlice passes the stabilizer pleteness. If every register of Ψ is in the state |Gi, then
test for g on ρ is [40] p = 1 + 1 for all i = 1,2,...,n. From the union
i test,i 2 2r+1
bound and the Hoeffding inequality,
1 I⊗n+s 1 Tr(ρg )
t i
p ≡ Tr ρ = + .
test,i 4r t∈{1X,2,3,4}r (cid:16) 2 (cid:17) 2 2r+1 Pr[Alice accepts Bob] = Pr n Ki ≥ 1 + 1−ǫ
k 2 2r+1
Verification protocol.— We now explain our verifica- hi^=1(cid:16) (cid:17)i
n
tion protocol. Bob sends Alice an n(nk+1+m)-qubit Ki 1 1−ǫ
≥ 1− Pr < +
state Ψ, where k = 22r+3n7 and m ≥ 2n7k2ln2. The Xi=1 h k 2 2r+1i
state Ψ consists of nk+1+m registers (Fig. 1). Each
n
K ǫ
register stores n qubits. (If Bob is honest, every regis- = 1− Pr i <p −
k test,i 2r+1
ter is in the state |Gi. If Bob is malicious, on the other Xi=1 h i
Ahalincde,rΨancdaonmbleyapneyrmnu(nteks+re1g+ismte)r-sqaunbditdeinstcaanrgdlsedmstraetgeis.)- ≥ 1−ne−222ǫr2+2k.
ters. (As we will see later, this randompermutation and
Proofofthesoundness.—Wenextshowthesoundness.
discarding of some registers are necessary to guarantee
We define the n-qubit projection operator Π⊥ ≡ I⊗n −
that the remaining state is close to an i.i.d. sample by G
|GihG|. Let T be the POVM element corresponding to
using the quantum de Finetti theorem [41].) Let Ψ′ be
the event that Alice accepts Bob. We can show that for
the remaining state. The state Ψ′ consists of nk+1 reg-
any n-qubit state ρ,
isters. She chooses one register from Ψ′, which is used
forthemeasurement-basedquantumcomputing. Wecall 1
Tr (T ⊗Π⊥)ρ⊗nk+1 ≤ . (1)
the registercomputing register. The remainingnk regis- G 2n2
h i
tersofΨ′ aredividedinton groups. Eachgroupconsists
Its proof is given later. Due to the quantum de Finetti
of k registers. The stabilizer test for g is performed on
i
theorem (for the one-way LOCC norm version) [41],
every register in the ith group for i = 1,2,...,n. (Note
that Alice does not need to do the permutation “phys-
ically”, which requires a quantum memory. Bob just Tr (T ⊗Π⊥)Ψ′ ≤ Tr (T ⊗Π⊥) dµ(ρ)ρ⊗nk+1
G G
sends each qubit of Ψ one by one to Alice, and Alice h i h Z i
randomly chooses her action from the test, discarding, 1 2n2k2nln2
+
or computation.) 2r m
Let Ki be the number of times that Alice passes the 1 1 2n3k2ln2 1
stabilizer test for g , i.e. the random variable to de- ≤ + = .
i 2n2 2r2n7k2ln2 n2
scribe the number of Alice’s observation of the event
41r t I⊗n2+st. If Kki ≥ 12 + 21r−+ǫ1, we say that the ith We have Tr[(T ⊗Π⊥G)Ψ′] = Tr(Π⊥Gρcomp)Tr[(T ⊗I)Ψ′].
groPup passes the test. Here, ǫ = 1 . If all groups pass Therefore, if Tr(Π⊥ρ )> 1, then Tr[(T ⊗I)Ψ′]< 1,
2n3 G comp n n
the test, we say that Alice accepts Bob. which means that if Alice accepts Bob, hG|ρ |Gi ≥
comp
1− 1 with a probability larger than 1− 1.
n n
Proof of Eq. (1).— First,letus assume thatTr(ρg )≥
i
1−δ foralli=1,2,..,n,whereδ = 1 . Duetotheunion
n3
bound,
test for g1 test for g2 test for g3 MBQC n I⊗n+gi
1−hG|ρ|Gi = 1−Tr ρ
2
(cid:16)iY=1 (cid:17)
FIG. 1: An example for n = 3, k = 2, m = 5. Each n I⊗n+g nδ
square represents a register that stores n qubits. Registers ≤ 1−Tr ρ i ≤ .
2 2
represented by black squaresare discarded. Xi=1h (cid:16) (cid:17)i
Therefore,
Themainresultsofthepresentpaperarethefollowing
two items:
Tr (T ⊗Π⊥)ρ⊗nk+1 = Tr(Tρ⊗nk)Tr(Π⊥ρ)
1. Completeness: if every register of Ψ is in the state G G
h i
|Gi,thentheprobabilitythatAlice acceptsBobislarger nδ 1
than 1−ne−n. ≤ 1× = . (2)
2 2n2
4
Next let us assume that Tr(ρg ) < 1−δ for at least where p′ is the output probability distribution of the
i z
one i. In this case, single-qubit measurements on ρ . Then,
comp
1 Tr(ρg ) 1 1−δ
ptest,i = 2 + 2r+1i < 2 + 2r+1 |pz −qz| ≤ |pz −p′z|+ |p′z −qz|
Xz Xz Xz
1
for the i. Then, due to the Hoeffding inequality, ≤ o(1)+ ,
192
Tr[(T ⊗I)ρ⊗nk+1] ≤ Pr[group i passes the test]
which means that the classical sampler can also sample
K 1 1−ǫ
= Pr i ≥ + pz with the ∼ 1/192 L1-norm error. For example, the
k 2 2r+1
h i hypergraph states naturally induced from the IQP cir-
K 1 1−δ δ−ǫ
= Pr i ≥ + + cuitscorrespondingtothenon-adaptiveUnionJackstate
k 2 2r+1 2r+1
h i measurement-basedquantumcomputing[39]canbeused
K δ−ǫ
≤ Pr i >p + for that purpose. Since the non-adaptive Union Jack
k test,i 2r+1
h i state measurement-based quantum computing is univer-
≤ e−2(2δ2−r+ǫ)22k =e−n. salwith postselections, a multiplicative errorcalculation
of its output probability distribution is #P-hard [20]. If
Hence we assume the worst case hardness can be lifted to the
average case one, we can show the hardness of the clas-
Tr[(T ⊗Π⊥)ρ⊗nk+1] = Tr(Tρ⊗nk)Tr(Π⊥ρ) sical constant L1-normerror sampling.
G G
≤ e−n×1. (3) TM is supported by the JST ACT-I, the JSPS Grant-
in-Aid for Young Scientists (B) No.26730003, and the
From Eqs. (2) and (3), for any state ρ, MEXT JSPS Grant-in-Aid for Scientific Research on
Innovative Areas No.15H00850. YT is supported by
1 1 the Program for Leading Graduate Schools: Interactive
Tr[(T ⊗Π⊥)ρ⊗nk+1]≤max ,e−n = .
G 2n2 2n2 Materials Science Cadet Program. MH is supported
(cid:16) (cid:17)
in part by Fund for the Promotion of Joint Interna-
Applications.— To conclude this paper, we finally dis- tional Research (Fostering Joint International Research)
cuss two applications of our results. First, our verifi- No.15KK0007, the JSPS MEXT Grant-in-Aid for Sci-
cation protocol can be used in verified blind quantum entific Research (B) No.16KT0017,the Okawa Research
computing. In the protocol of Ref. [28], the user needs Grant and Kayamori Foundation of Information Science
non-Clifford basis measurements to implement quantum Advancement.
computing (the verification itself can be done with only
Pauli measurements.) On the other hand, if the server
generates the Union Jack states [39], for example, the
user needs only Pauli measurements for both the verifi-
cation and the computation. ∗ Electronic address: [email protected]
† Electronic address: [email protected]
Second,ourprotocolcanbeusedfortheverifiedquan-
‡ Electronic address: [email protected]
tumsupremacydemonstration. ItwasshowninRef.[19]
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