6.7 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| BQP | 3/3 | https://en.wikipedia.org/wiki/BQP | reference | science, encyclopedia | 2026-05-05T11:06:41.292100+00:00 | kb-cron |
Consider a quantum circuit C, which consists of t gates,
g
1
,
g
2
,
⋯
,
g
m
{\displaystyle g_{1},g_{2},\cdots ,g_{m}}
, where each
g
j
{\displaystyle g_{j}}
comes from a universal gate set and acts on at most two qubits. To understand what the sum of histories is, we visualize the evolution of a quantum state given a quantum circuit as a tree. The root is the input
|
0
⟩
⊗
n
{\displaystyle |0\rangle ^{\otimes n}}
, and each node in the tree has
2
n
{\displaystyle 2^{n}}
children, each representing a state in
C
n
{\displaystyle \mathbb {C} ^{n}}
. The weight on a tree edge from a node in j-th level representing a state
|
x
⟩
{\displaystyle |x\rangle }
to a node in
j
+
1
{\displaystyle j+1}
-th level representing a state
|
y
⟩
{\displaystyle |y\rangle }
is
⟨
y
|
g
j
+
1
|
x
⟩
{\displaystyle \langle y|g_{j+1}|x\rangle }
, the amplitude of
|
y
⟩
{\displaystyle |y\rangle }
after applying
g
j
+
1
{\displaystyle g_{j+1}}
on
|
x
⟩
{\displaystyle |x\rangle }
. The transition amplitude of a root-to-leaf path is the product of all the weights on the edges along the path. To get the probability of the final state being
|
ψ
⟩
{\displaystyle |\psi \rangle }
, we sum up the amplitudes of all root-to-leave paths that ends at a node representing
|
ψ
⟩
{\displaystyle |\psi \rangle }
. More formally, for the quantum circuit C, its sum over histories tree is a tree of depth m, with one level for each gate
g
i
{\displaystyle g_{i}}
in addition to the root, and with branching factor
2
n
{\displaystyle 2^{n}}
.
Notice in the sum over histories algorithm to compute some amplitude
α
x
{\displaystyle \alpha _{x}}
, only one history is stored at any point in the computation. Hence, the sum over histories algorithm uses
O
(
n
m
)
{\displaystyle O(nm)}
space to compute
α
x
{\displaystyle \alpha _{x}}
for any x since
O
(
n
m
)
{\displaystyle O(nm)}
bits are needed to store the histories in addition to some workspace variables. Therefore, in polynomial space, we may compute
∑
x
|
α
x
|
2
{\displaystyle \sum _{x}|\alpha _{x}|^{2}}
over all x with the first qubit being 1, which is the probability that the first qubit is measured to be 1 by the end of the circuit. Notice that compared with the simulation given for the proof that
B
Q
P
⊆
E
X
P
{\displaystyle {\mathsf {BQP}}\subseteq {\mathsf {EXP}}}
, our algorithm here takes far less space but far more time instead. In fact it takes
O
(
m
⋅
2
m
n
)
{\displaystyle O(m\cdot 2^{mn})}
time to calculate a single amplitude!
=== BQP and PP === A similar sum-over-histories argument can be used to show that
B
Q
P
⊆
P
P
{\displaystyle {\mathsf {BQP}}\subseteq {\mathsf {PP}}}
.
=== P and BQP === We know
P
⊆
B
Q
P
{\displaystyle {\mathsf {P}}\subseteq {\mathsf {BQP}}}
, since every classical circuit can be simulated by a quantum circuit. It is conjectured that BQP solves hard problems outside of P, specifically, problems in NP. The claim is indefinite because we don't know if P=NP, so we don't know if those problems are actually in P. Below are some evidence of the conjecture:
Integer factorization (see Shor's algorithm) Discrete logarithm Simulation of quantum systems (see universal quantum simulator) Approximating the Jones polynomial at certain roots of unity Harrow-Hassidim-Lloyd (HHL) algorithm
== See also == Hidden subgroup problem Polynomial hierarchy (PH) Quantum complexity theory QMA, the quantum equivalent to NP. QIP, the quantum equivalent to IP.
== References ==
== External links == Complexity Zoo link to BQP Archived 2013-06-03 at the Wayback Machine