System and method for performing information—theoretically secure quantum gate computation and quantum key distribution, based on random rotation of qubits
View Patent ↗A computer implemented method for encoding bits by qubits to perform information-theoretically secure quantum gate computation, according to which pairs of quantum bits consisting of a first qubit as an encoding of “0” and a second qubit as an encoding of “1” are randomly selected, such that the first and second qubits are orthogonal to each other as quantum states and are interchanged by a NOT gate. Each qubit rotating to a desired initial direction and then each rotated qubit is further rotated to its antipodal direction by applying a quantum NOT or CNOT gate to the each rotated qubit, without any knowledge about the desired direction. A unitary gate is further applied over the qubits, using an ancillary |0 qubit that creates an equally weighted superposition of the qubits.
1. A computer implemented method for distributing a quantum key for performing information-theoretically secure quantum gate computation, comprising:
a) generating, by at least one processor, a quantum key k=(θ, φ) by uniformly selecting a random pair (θ, φ) from
[
0
,
2
π
]
×
{
π
2
,
-
π
2
}
;
b) setting, by said at least one processor, an element
ψ
0
=
cos
(
θ
2
)
+
e
i
φ
sin
(
θ
/
2
)
,
as an encryption of “0” using the quantum key k=(θ, φ) as the encryption key;
c) setting, by said at least one processor, an element ψ 1 =sin(θ/2)−e iφ cos(θ/2) to be the encryption of “1” using said key, such that ψ 0 and ψ 1 are orthogonal;
d) constituting, by said at least one processor, a random orthonormal basis B (θ,φ) ; and
e) allowing ψ 0 to be equal to NOTψ 1 and vice versa by applying a NOT operator executed by a NOT gate to ψ 1 and selecting φ=±π/2, such that said random orthonormal basis is
B
(
θ
,
±
π
2
)
=
{
cos
(
θ
2
)
±
i
sin
(
θ
2
)
,
sin
(
θ
2
)
∓
i
cos
(
θ
2
)
}
;
f) transmitting, by a transceiver coupled to said at least one processor, the generated quantum key.
2. A method according to claim 1 , further comprising using elements ψ 0 and ψ 1 or bit-wise encryption of a string x of classical bits.
3. A method according to claim 1 , wherein encryption operation on an input message b∈M, where is a message space, is done using the key k=(θ, φ), by:
a) generating qubit |b ;
b) applying K to |b to obtain |q =K|b , where
K
=
(
cos
(
θ
/
2
)
sin
(
θ
/
2
)
e
i
φ
sin
(
θ
/
2
)
-
e
i
φ
cos
(
θ
/
2
)
)
∈
M
2
(
ℂ
)
;
and
c) outputting an encrypted message |q .
4. A method according to claim 1 , wherein decryption operation on input ciphertext |ψ is done using the quantum key k=(θ, φ), by:
a) applying K † being a conjugate transpose of
K
=
(
cos
(
θ
/
2
)
sin
(
θ
/
2
)
e
i
φ
sin
(
θ
/
2
)
-
e
i
φ
cos
(
θ
/
2
)
)
∈
M
2
(
ℂ
)
to |ψ , to obtain K † |ψ ;
b) measuring K † |ψ with reference to a computational basis
B
(
θ
,
±
2
π
¯
)
=
{
cos
(
θ
/
2
)
±
i
sin
(
θ
/
2
)
,
sin
(
θ
/
2
)
∓
i
cos
(
θ
/
2
)
}
;
and
c) outputting a measurement result.
5. A method according to claim 1 , further comprising performing IT-secure quantum gate computations over encrypted data comprising at least one of the encryption of “0” or the encryption of “1”.
6. A method according to claim 5 , wherein the quantum gate is selected from the group of:
NOT gates;
a quantum gate, using an ancillary |0> qubit, that takes elements of the orthonormal basis to equally weighted superpositions of said elements;
controlled-NOT (CNOT) and C n NOT gates, where control qubits are set as plaintext qubits.
7. A method according to claim 1 , wherein any gate that commutes with a family of unitary matrices K, defined in |Enc|, is applied homomorphically to encrypted data comprising at least one of the encryption of “0” or the encryption of “1”.
8. A computer being capable of distributing a quantum key for performing information-theoretically secure quantum gate computation, comprising at least one processor and a transceiver coupled to said at least one processor, adapted to:
a) generate a quantum key k=(θ,φ) by selecting a uniformly random pair (θ, φ) from
[
0
,
2
π
]
×
{
π
2
,
-
π
2
}
;
b) set an element
ψ
0
=
cos
(
θ
2
)
+
e
i
φ
sin
(
θ
/
2
)
,
an encryption of “0” using the quantum key k=(θ,φ) as the encryption key;
c) set an element ψ 1 =sin(θ/2)−e iφ cos(θ/2) to be the encryption of “1” using said key, such that ψ 0 and ψ 1 are orthogonal;
d) constitute a random orthonormal basis B (θ,φ) ;
e) select φ=±π/2, such that said random orthonormal basis is
B
(
θ
,
±
π
2
)
=
{
cos
(
θ
/
2
)
±
i
sin
(
θ
/
2
)
,
sin
(
θ
/
2
)
∓
i
cos
(
θ
/
2
)
}
;
f) set ψ 0 to be equal to NOTψ 1 and vice versa by applying a NOT operator executed by a NOT gate to ψ 1 ; and
g) transmit the generated quantum key by said transceiver.
9. A computer according to claim 8 , further adapted to perform IT-secure quantum gate computations over encrypted data comprising at least one of the encryption of “0” or the encryption of “1”.
10. A computer according to claim 8 , wherein the quantum gate is selected from the group of:
NOT gates;
a quantum gate, using an ancillary |0> qubit, that takes elements of the orthonormal basis to equally weighted superpositions of said elements;
controlled-NOT (CNOT) and C n NOT gates, where control qubits are set as plaintext qubits.
11. A method according to claim 1 , wherein both “0” and “1” are encoded as the same qubit.