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G(cid:48)(|φ i(cid:105)⊗|ϕ(cid:105))=|φ(cid:48) i(cid:105)⊗U i|ϕ(cid:105), (1) quantum gate sequence of U i. If the key is stolen, there |
isnowaytopreventEvefromperformingU ,butweneed |
where |φ(cid:48)(cid:105) = L|φ (cid:105) is the key state after performing G(cid:48). i |
i i topreventEvecreatinganunauthorizedcopyofthekey. |
The programmer issues only one key for each authorized |
We assume that Eve may destroy extra keys {|φ (cid:105)} for |
user. This step is the encryption process of the keys. j |
j (cid:54)=i,butdoesnotdestroythekey|φ (cid:105)forperformingU . |
Step3: Theessenceofourobfuscationprocessisinthe i i |
Weregardthatthekeyshouldbekeptastheevidenceof |
non-uniqueness of the quantum gate sequence represen- |
the authorized user. |
tation for unitary operations. The programmer trans- |
Thus, we formally define authorized quantum com- |
forms the series of quantum gate sequences x(G(cid:48)) = |
putation implementing 2k unitary operations {U } on |
x(R)x(G)x(L)intoanotherobfuscatedquantumgatese- i |
H⊗n by the existence of a quantum gate sequence |
quence x(cid:48)(G(cid:48)), where extracting information of R and L input |
x(cid:48)(G(cid:48)) and quantum authorization keys {|φ (cid:105)} satisfying |
from x(cid:48)(G(cid:48)) is not possible in a polynomial time even i |
thefollowingtwoconditions. 1. G(cid:48)isaunitaryoperation |
using quantum computers. We call this transformation |
onanextendedHilbertspaceH⊗(m+n) satisfyingEq.(1) |
of quantum gate sequences as quantum gate shuffling. |
for an arbitrary input state |ϕ(cid:105)∈H⊗n . 2. There is no |
Later, we present a sufficient condition for a quantum input |
polynomial quantum algorithm A such that |
gate shuffling algorithm for performing the obfuscation |
process. A:(x(cid:48)(G(cid:48)), |φ (cid:105)⊗|Φ (cid:105))(cid:55)−→(y(F(cid:48)), |φ (cid:105)⊗|ψy (cid:105)) (2) |
Step4: Classicalinformationoftheobfuscatedgatese- i iN i x(cid:48)i |
quencex(cid:48)(G(cid:48))isdeliveredtothetrustedthirdparty(au- |
where |Φ (cid:105)=|φ (cid:105)⊗···⊗|φ (cid:105) is a product state of N |
thenticator) by the programmer. It is authenticated and keys (i i =N j ···jj1 ∈{0,1}kNjN ) and y(F(cid:48)) is a quantum |
N 1 N |
announced publicly. On the other hand, the program- gate sequence of a unitary operation F(cid:48), which allows U |
i |
mer directly sends the key |φ i(cid:105) to an authorized user via to be performed by using a forged key state |ψy (cid:105)∈H⊗l |
x(cid:48)i |
a quantum channel. A pair of an authenticated public (for some integer l) as |
programx(cid:48) andasetofauthorizationprivatekeys{|φ (cid:105)} |
i |
is created. F(cid:48)(|ψ xy (cid:48)i(cid:105)⊗|ϕ(cid:105))=(cid:12) (cid:12)ψ x(cid:48)y (cid:48)i(cid:11) ⊗U i|ϕ(cid:105), (3) |
Step 5: The user performs a unitary operation G(cid:48) de- |
scribed by x(cid:48)(G(cid:48)) on the joint state of the key |φ (cid:105) and for an arbitrary |ϕ(cid:105)∈H⊗n and (cid:104)ψy |ψy (cid:105)=δ . |
i input x(cid:48)i x(cid:48)j ij |
an input state |ϕ(cid:105) of user’s choice. Then the state U |ϕ(cid:105) Next,weinvestigatequantumgateshufflingalgorithms |
i |
is obtained. If the user does not use the correct key |φ (cid:105) for the obfuscation process. Among algorithms mapping |
i |
and performs G(cid:48), the resulting joint state cannot be a an element of g (U) to another element of g (U) |
p(n) q(n) |
3 |
where p(n) ≤ q(n), we define a completely-random shuf- we require q(n) to be just a polynomial function. |
fling to be an algorithm randomly obtaining a quantum By assuming the existence of the sufficiently-random |
gate sequence from all possible quantum gate sequences gate shuffling algorithm, we prove that it is quantum- |
of g (U). To understand the power of random shuf- computationally difficult for Eve to perform a cracking |
q(n) |
fling, we study restricted quantum gate sequences de- algorithmAdefinedbyEq.(2). Weshowthatthequan- |
noted by z(C ) of a controlled identity operation C on tum computational complexity of this task is in NQP |
I I |
H⊗(n+1) constructed by a quantum gate sequence x(I). (Non-deterministic Quantum Polynomial)-hard class by |
Foragivenpolynomialquantumgatesequencex(U)ofa reducing it to a NQP-Complete problem, the exact non- |
generalunitaryoperationU onH⊗n, wecanalwayscon- identitycheckproblem[6]oflargeunitarygatesequences. |
struct a corresponding quantum gate sequence z(C ) of The exact non-identity check problem is defined by the |
U |
acontrolledunitaryoperationC onH⊗(n+1) byadding following. Letxbeaquantumgatesequenceimplement- |
U |
a control qubit in front of original qubits, replacing all ing a unitary operation U with an ancilla system, decide |
the gate elements by controlled-gate operations and fur- whether U is proportional to the identity operation, i.e., |
ther decomposing them into elementary gate operations. U = eiθI, or not. It is proven in Ref. [6] that computa- |
This procedure can be completed in polynomial steps in tionalcomplexityoftheexactnon-identitycheckproblem |
|x(U)|. Note that the restricted quantum gate sequence isNQP-Complete[7]. TheclassNQPisconsideredtobe |
z(C ) can be also constructed from x(eiθI).[5] oneofthenaturalextensionsoftheclassNPtoquantum |
I |
We consider that a quantum gate sequence x(I) ∈ computational complexity. |
g (I) is given by a non-trivial combination of elemen- To apply the algorithm A to the exact non-identity |
p(n) |
tarygatesandwefurtherapplyaquantumgatesequence check problem, we introduce a modified non-identity |
ofaunitaryoperationV actingonlyonthecontrolqubit check problem by extending a quantum gate sequence |
(the first qubit) Hilbert space H . We compare the x(U) on H⊗n into a restricted quantum gate sequence |
control |
quantum gate sequences of (V ⊗I)C and C (V ⊗I). z(C ) on H⊗(n+1). Then we apply two unitary opera- |
I I U |
If there exists a random shuffling algorithm in polyno- tions V and V on H (the controlled qubit) from the |
L R |
mial time, both sets are given by g(V ⊗I) and they are lefthandsideandtherighthandsideof C . Theresult- |
U |
identical. Thus, after the random shuffling process, we ing operation is written by C(cid:48) = (V ⊗I)C (V ⊗I). |
U L U R |
cannot distinguish whether the quantum gate sequence SimilarlytoEq.(1),thisoperationtransformsC(cid:48) (|φ (cid:105)⊗ |
U i |
of V was originally applied from the right-hand side of |ϕ(cid:105)) = |φ(cid:48)(cid:105)⊗Ui|ϕ(cid:105) for an arbitrary input state |ϕ(cid:105) ∈ |
i |
the controlled identity or from the left-hand side. Infor- H⊗n, where i ∈ {0,1}, |φ (cid:105) = V †|i(cid:105) and |φ(cid:48)(cid:105) = V |i(cid:105) |
i R i L |
mationof theposition ofV is lost. This informationloss forasingle-qubitkeystate. (NotethatUidenotestheith |
is a key idea for our security proof. power U and it is different from U .) We state the mod- |
i |
However, the existence of a completely-random gate ified non-identity check problem as the following: Given |
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