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Insteadofthesingle-qubitstategeneration,theability |
Bob touches some traps. |
of single-qubit measurements is also enough for Alice: it |
Recently,anew verificationprotocolthatdoesnotuse |
wasshowninRef.[15]thatAlicewhocandoonlysingle- |
the traptechnique was proposed[22] (see alsoRef. [23]). |
qubitmeasurementscanperformblindquantumcomput- |
In this protocol, Bob generates a graph state, and sends |
ing. The idea is that Bob generates a graph state and |
each qubit of it one by one to Alice. Alice directly veri- |
sendseachqubitonebyonetoAlice. IfBobishonest,he |
fies the correctness of the graph state (and therefore the |
generatesthecorrectgraphstateandthereforeAlice can |
correctness of the computation) sent from Bob by mea- |
perform the correct measurement-based quantum com- |
suringstabilizer operators. This verificationtechnique is |
puting (correctness). If Bob is malicious, he might send |
called the stabilizer test. Note that the stabilizer test is |
a wrongstate to Alice, but whateverBobsends to Alice, |
useful also in quantum interactive proof system [24–28]. |
Alice’s measurement angles, which contain information |
aboutAlice’scomputation,cannotbetransmittedtoBob Althoughcomputingitselfisverifiablethroughthesta- |
due to the no-signalingprinciple (blindness). The proto- bilizertest,theinputisnotifitisaquantumstatewhose |
col is called the measurement-only protocol, since Alice classical efficient description is not known to Alice. For |
needs only measurements. example, let us assume that Alice receives a state ψ |
| i |
A problem in all these protocols is the lack of the ver- fromCharlie,andshewantstoapplyaunitaryU on ψ . |
| i |
If she delegates the quantum computation to Bob in the |
measurement-only style, a possible procedure is as fol- |
lows. Alicefirstsends ψ toBob. Bobnextentangles ψ |
| i | i |
∗Electronicaddress: morimae@gunma-u.ac.jp tothegraphstate. Bobthensendseachqubitofthestate |
2 |
one by one to Alice, and Alice does measurement-based In fact, if p 1 ǫ, we obtain |
pass |
≥ − |
quantum computing on it. If Bob is honest, Alice can |
realize U ψ . Furthermore, as is shown in Refs. [24], Al- k I +g |
| i Tr j ρ 1 2ǫ. |
ice can verify the correctness of the graphstate by using |
the stabilizer test even if some states are coupled to the (cid:16) jY=1 2 (cid:17)≥ − |
graph state. However, in the procedure, the correctness |
Let |
of the input state is not guaranteed, since Bob does not |
necessarily couple ψ to the graph state, and Alice can- k |
| i I+g j |
not check the correctness of the input state. Bob might Λ . |
≡ 2 |
discard ψ and entangles completely different state ψ′ jY=1 |
| i | i |
to the graph state. In this case what Alice obtains is |
not U ψ but U ψ′ . Can Alice verify that Bob honestly From the gentle measurement lemma [29], |
| i | i |
coupled her input state to the graph state? |
1 |
In this paper, we introduce a new protocol of ρ ΛρΛ 1 Tr(Λρ) |
1 |
2k − k ≤ − |
measurement-only verifiable blind quantum computing p |
where notonlythe computationitself but alsothe quan- 1 (1 2ǫ) |
≤ − − |
tum input are verifiable. Our strategy is to combine the = p√2ǫ. |
traptechniqueandthestabilizertest. Thecorrectnessof |
the computing is verified by the stabilizer test, and the Note that |
correctness of the quantum input is verified by checking |
ΛρΛ ΛρΛ |
the trap qubits that are randomly hidden in the input g g = |
j j |
Tr(Λρ) Tr(Λρ) |
state. When the traps are checked, the state can be iso- |
lated from the graph state by measuring the connecting |
foranyj,andtherefore,ΛρΛ/Tr(Λρ)isastabilizedstate. |
qubits in Z basis. The main technical challenge in our |
For any positive operator M, |
proof is to show that the trap verification and the stabi- |
lizer verification can coexist with each other. Tr(Mρ) Tr(ΛρΛ) √2ǫ, |
− ≤ |
which means |
II. STABILIZER TEST |
ΛρΛ |
Tr(Mρ) Tr M Tr(Λρ)+√2ǫ |
We first review the stabilizer test. Let us consider an ≤ (cid:16) Tr(Λρ) (cid:17) |
N-qubitstateρandasetg ≡{g 1,...,g n }ofgeneratorsof Tr M ΛρΛ +√2ǫ. |
a stabilizer group. The stabilizer test is a following test: ≤ (cid:16) Tr(Λρ) (cid:17) |
1. Randomly generate an n-bit string k And, for any positive operator M, |
(k ,...,k ) 0,1 n. ≡ |
1 n |
∈{ } Tr(ΛρΛ) Tr(Mρ) √2ǫ, |
2. Measure the operator − ≤ |
which means |
n |
jkj. |
s k ≡ g Tr(Mρ) Tr M ΛρΛ Tr(Λρ) √2ǫ |
jY=1 ≥ (cid:16) Tr(Λρ) (cid:17) − |
ΛρΛ |
Note that this measurement can be done with Tr M (1 2ǫ) √2ǫ. |
single-qubit measurements, since s k is a tensor ≥ (cid:16) Tr(Λρ) (cid:17) − − |
product of Pauli operators. |
3. If the result is +1 ( 1), the test passes (fails). III. OUR PROTOCOL |
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