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quantum

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(Dated: October 28, 2021)
We present authorized quantum computation, where only a user with a non-cloneable quantum
authorization key can perform a unitary operation created by an authenticated programmer. The
9002 securityofourauthorizedquantumcomputationisbasedonthequantumcomputationalcomplexity
problemofforgingthekeysfromanobfuscatedquantumgatesequence. Undertheassumptionofthe
existence of a sufficiently-random gate shuffling algorithm, the problem is shown to be in the NQP
(Non-deterministic Quantum Polynomial)-hard class by reducing it to a NQP-Complete problem,
the exact non-identity check problem. Therefore, our authorized quantum computation can be
raM
computationally secure against attacks using quantum computers.
PACSnumbers: 03.67.Ac,03.67.Lx,03.67.Dd
21
Considertheworldoncequantumcomputersexistand weshowasufficientconditionfortheobfuscationprocess
]hp-tnauq[ arewidelyused. Inthisworld,unitaryoperationsarethe where forging the quantum authorization keys is compu-
programsofquantumcomputers. Fortheprogrammerof tationally difficult even using quantum computers. We
the quantum programs, they are important intellectual stressthatwedonotobfuscatethequantumprogramit-
properties and it is important to protect the copyright self, but the identity of the quantum authorization keys.
of the programs. On the other hand, for the user of the
We note that if we do not require the authenticity
program, they do not need to know about the details of
of the program, blind quantum computation [3], which
the program, they just want to perform a task, as long
aims to perform a unitary operation without revealing
as the created by the authenticated programmer.
the identity of input states, can be used for similar
1v8802.3090:viXra Such a situation can be solved if the programmer en- tasks. Blind quantum computation is a two-party proto-
codes a program so that the original program is only colbasedoninformationalsecurityanditrequiresmulti-
performable for users with non-cloneable authorization ple quantum/classical communications during computa-
keys, distributes the encoded programs via an authen- tion. In contrast, our authorized quantum computation
ticator, and sends the authorization keys directly to the is a semi-public protocol based on computational secu-
users. Then,anyonecandownloadtheencodedprograms rity to ensure the authenticity of the program and no
viatheauthenticator,whichareguaranteedtohavebeen communication is required during computation.
made by the programmer, but it is performable only by
We first present a construction of authorized quantum
theauthorizeduser. Inthisletter,weproposeaschemeof
computation to sketch our scheme. We regard unitary
authorizedquantumcomputationthatallowsthistaskin
operations to be quantum programs. Note that we con-
quantumcomputationalsecurity,asapossiblenewquan-
sider only polynomial quantum programs, namely, uni-
tum cryptographic primitive.
tary operations represented by an array of a polyno-
In our scheme, we employ both quantum advantages mial number p(n) of elementary unitary gates. A uni-
and quantum limitations for its security. To protect the tary operation U is described by a polynomial classi-
originalprogramsfromunauthorizedusers, theencoding cal bit sequence {0,1}∗. We call this classical informa-
process of the program is two-fold. One is an encryp- tion as a quantum gate sequence of U and denote it by
tion process of introducing quantum authorization keys x(U). (Throughout this letter, we use capital letters to
so that computation is not possible without using the represent unitary operations and small letters to repre-
correct key. The keys are unknown quantum states for sent their quantum gate sequences.) Due to the non-
anyusers(evenforauthorizedusers)andtheiranonymity uniqueness of the gate sequence representation for uni-
and non-cloneability is ensured by quantum mechanics. tary operations, we consider a set of all quantum gate
The other is an obfuscation process that hides the basis sequencesforaunitaryoperationU consistingofatmost
of the keys in the encoded program. It is known that p(n) elementary gate arrays and denote it by g (U),
p(n)
classically, obfuscating programs is impossible [1]. For orsimplyg(U)ifthespecificationofafunctionp(n)isir-
quantum settings, existence of obfuscation with the help relevant. Since calculations of the matrix representation
of quantum states is an open problem [2]. In this letter, ofaunitaryoperationrequireexponentialcomputational
by introducing the concept of gate shuffling algorithms, power in terms of n, we have to rely on the polynomial
2
gate sequence representation of unitary operations when
n is large. Now consider a programmer who wants to
encode a unitary operation U (where i is an index for
i
specifying the unitary operation) acting on a n-qubit in-
put Hilbert space H⊗n .
input
Step 1: The programmer extends the U into another
i
unitary operation G acting on a larger Hilbert space
H⊗(m+n) by adding a m-qubit Hilbert space H⊗m of
key
quantum authorization keys in front of the input Hilbert FIG. 1: A construction of gate sequences for the unitary op-
space. (We often simply denote the quantum authoriza- erations G and G(cid:48). See the text for notations.
tion key as the key.) This extension is similar to the
programmablequantumgatearraysproposedbyNielsen
desired state and is highly likely to be entangled. The
andChuang[4]. TheextendedunitaryoperatorGtrans-
security of our scheme will be discussed later.
forms G(|i(cid:105)⊗|ϕ(cid:105)) = |i(cid:105)⊗U |ϕ(cid:105), where {U } is a set of
i i
P =2k (1≤i≤2k (cid:28)2m) unitary operations for an ar- We note that by performing the reverse quantum gate
bitrary input state |ϕ(cid:105)∈H⊗n , and {|i(cid:105)∈H⊗m} is the sequenceusingtheusedkeystate|φ(cid:48) i(cid:105),thereverseunitary
corresponding key states inin apu cot mputational bk ae sy is speci- operation U† can be also performed and the original key
i
fied by a binary bit sequence {0,1}k. The number of the |φ (cid:105)isregainedbyG(cid:48)†(|φ(cid:48)(cid:105)⊗|ϕ(cid:105))=|φ (cid:105)⊗U†|ϕ(cid:105). There-
i i i i
key qubits k should be taken to be of order log(n) for fore, we can recycle the quantum authorization key as
restricting the total gate number of G to be in polyno- long as it keeps coherence.
mialofn. Thusthe(m−k)-qubitdummyspaceH⊗(m−k) In this scheme, security depends on the obfuscation
dummy
is introduced in the key space. A construction of G for process given in Step 3. To present a sufficient condition
{U } is shown in Fig. 1, where M and M are random for obfuscation, we investigate strategies for a malicious
i 1 2
unitary operations acting on the dummy qubit space. user, Eve. We consider that Eve wants to perform the
Step 2: By applying random unitaryoperations L and originalunitaryoperationU i withoutusingthequantum
R on only H⊗m as G → G(cid:48) = (L⊗I)G(R⊗I) where I authorizationkey|φ i(cid:105)issuedbytheprogrammer. Apow-
key
denotesanidentityoperatorofanappropriatedimension, erful Eve may also be able to tap the quantum channels
we create a key state |φ i(cid:105)=R†|i(cid:105) satisfying and steal other N keys {|φ j(cid:105)} j=1,...,N where j (cid:54)= i, and
analyze the quantum gate sequence x(cid:48)(G(cid:48)) to obtain a