8.9 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| BQP | 2/3 | https://en.wikipedia.org/wiki/BQP | reference | science, encyclopedia | 2026-05-05T11:06:41.292100+00:00 | kb-cron |
==== APPROX-QCIRCUIT-PROB ==== The APPROX-QCIRCUIT-PROB problem is complete for efficient quantum computation, and the version presented below is complete for the Promise-BQP complexity class (and not for the total BQP complexity class, for which no complete problems are known). APPROX-QCIRCUIT-PROB's completeness makes it useful for proofs showing the relationships between other complexity classes and BQP. Given a description of a quantum circuit C acting on n qubits with m gates, where m is a polynomial in n and each gate acts on one or two qubits, and two numbers
α
,
β
∈
[
0
,
1
]
,
α
>
β
{\displaystyle \alpha ,\beta \in [0,1],\alpha >\beta }
, distinguish between the following two cases:
measuring the first qubit of the state
C
|
0
⟩
⊗
n
{\displaystyle C|0\rangle ^{\otimes n}}
yields
|
1
⟩
{\displaystyle |1\rangle }
with probability
≥
α
{\displaystyle \geq \alpha }
measuring the first qubit of the state
C
|
0
⟩
⊗
n
{\displaystyle C|0\rangle ^{\otimes n}}
yields
|
1
⟩
{\displaystyle |1\rangle }
with probability
≤
β
{\displaystyle \leq \beta }
Here, there is a promise on the inputs as the problem does not specify the behavior if an instance is not covered by these two cases. Claim. Any BQP problem reduces to APPROX-QCIRCUIT-PROB. Proof. Suppose we have an algorithm A that solves APPROX-QCIRCUIT-PROB, i.e., given a quantum circuit C acting on n qubits, and two numbers
α
,
β
∈
[
0
,
1
]
,
α
>
β
{\displaystyle \alpha ,\beta \in [0,1],\alpha >\beta }
, A distinguishes between the above two cases. We can solve any problem in BQP with this oracle, by setting
α
=
2
/
3
,
β
=
1
/
3
{\displaystyle \alpha =2/3,\beta =1/3}
. For any
L
∈
B
Q
P
{\displaystyle L\in {\mathsf {BQP}}}
, there exists family of quantum circuits
{
Q
n
:
n
∈
N
}
{\displaystyle \{Q_{n}\colon n\in \mathbb {N} \}}
such that for all
n
∈
N
{\displaystyle n\in \mathbb {N} }
, a state
|
x
⟩
{\displaystyle |x\rangle }
of
n
{\displaystyle n}
qubits, if
x
∈
L
,
P
r
(
Q
n
(
|
x
⟩
)
=
1
)
≥
2
/
3
{\displaystyle x\in L,Pr(Q_{n}(|x\rangle )=1)\geq 2/3}
; else if
x
∉
L
,
P
r
(
Q
n
(
|
x
⟩
)
=
0
)
≥
2
/
3
{\displaystyle x\notin L,Pr(Q_{n}(|x\rangle )=0)\geq 2/3}
. Fix an input
|
x
⟩
{\displaystyle |x\rangle }
of n qubits, and the corresponding quantum circuit
Q
n
{\displaystyle Q_{n}}
. We can first construct a circuit
C
x
{\displaystyle C_{x}}
such that
C
x
|
0
⟩
⊗
n
=
|
x
⟩
{\displaystyle C_{x}|0\rangle ^{\otimes n}=|x\rangle }
. This can be done easily by hardwiring
|
x
⟩
{\displaystyle |x\rangle }
and apply a sequence of CNOT gates to flip the qubits. Then we can combine two circuits to get
C
′
=
Q
n
C
x
{\displaystyle C'=Q_{n}C_{x}}
, and now
C
′
|
0
⟩
⊗
n
=
Q
n
|
x
⟩
{\displaystyle C'|0\rangle ^{\otimes n}=Q_{n}|x\rangle }
. And finally, necessarily the results of
Q
n
{\displaystyle Q_{n}}
is obtained by measuring several qubits and apply some (classical) logic gates to them. We can always defer the measurement and reroute the circuits so that by measuring the first qubit of
C
′
|
0
⟩
⊗
n
=
Q
n
|
x
⟩
{\displaystyle C'|0\rangle ^{\otimes n}=Q_{n}|x\rangle }
, we get the output. This will be our circuit C, and we decide the membership of
x
∈
L
{\displaystyle x\in L}
by running
A
(
C
)
{\displaystyle A(C)}
with
α
=
2
/
3
,
β
=
1
/
3
{\displaystyle \alpha =2/3,\beta =1/3}
. By definition of BQP, we will either fall into the first case (acceptance), or the second case (rejection), so
L
∈
B
Q
P
{\displaystyle L\in {\mathsf {BQP}}}
reduces to APPROX-QCIRCUIT-PROB.
=== BQP and EXP === We begin with an easier containment. To show that
B
Q
P
⊆
E
X
P
{\displaystyle {\mathsf {BQP}}\subseteq {\mathsf {EXP}}}
, it suffices to show that APPROX-QCIRCUIT-PROB is in EXP since APPROX-QCIRCUIT-PROB is BQP-complete.
Note that this algorithm also requires
2
O
(
n
)
{\displaystyle 2^{O(n)}}
space to store the vectors and the matrices. We will show in the following section that we can improve upon the space complexity.
=== BQP and PSPACE === Sum of histories is a technique introduced by physicist Richard Feynman for path integral formulation. APPROX-QCIRCUIT-PROB can be formulated in the sum of histories technique to show that
B
Q
P
⊆
P
S
P
A
C
E
{\displaystyle {\mathsf {BQP}}\subseteq {\mathsf {PSPACE}}}
.