kb/data/en.wikipedia.org/wiki/BQP-2.md

---
title: "BQP"
chunk: 3/3
source: "https://en.wikipedia.org/wiki/BQP"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:06:41.292100+00:00"
instance: "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