Computer Science
Mathematics
Physics
Quantum Computation

Quantum Phase Estimation

Construct the quantum circuit to estimate the eigenphase (or phase) of a given eigenvector of a unitary operator

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The quantum phase estimation algorithm solves the problem of finding in the eigenvalue equation

Uψ=

2πθ



, where is

a unitary operator. The inputs of the algorithm are qubits at the initial state

⊗n

|0〉

, and the post-measurement state with largest probability is

θ〉

Install and load the QuantumFramework paclet:

In[1]:=

PacletInstall["Wolfram/QuantumFramework"]Needs["Wolfram`QuantumFramework`"]

Out[1]=

PacletObject

Name: Wolfram/QuantumFramework

Version: 1.0.32



The built-in circuit "PhaseEstimation" takes two input arguments: a unitary operator

and an integer

. For

, let's consider a phase operator.

For a phase operator with phase

, find eigenvalues:

In[2]:=

QuantumOperator[{"Phase",2πθ}]["Eigenvalues"]

Out[2]=

1,

2πθ





The integer

specifies the number of qubits and

controlled-

operators in the circuit, with

j=0,1,…,n-1

. The accuracy of phase estimation and the success probability depends on

(number of qubits):

In[3]:=

n=4;θ=1/5;qc=QuantumCircuitOperator[{"PhaseEstimation",QuantumOperator[{"Phase",2πθ}],n}];qc["Diagram",FontSize11]

Out[3]=

Return the corresponding measurement, with all qubits prepared in state:

In[4]:=

mea=N@qc[]

Out[4]=

QuantumMeasurement

Target: {1,2,3,4}

Measurement Outcomes: 16



Plot corresponding probabilities:

In[5]:=

mea["ProbabilityPlot"]

Out[5]=

Given the outcome with the largest probability, estimate the phase:

In[6]:=

FromDigits[Keys[TakeLargest[mea["Probabilities"],1]]〚1〛["Name"],2]/2^n//N

Out[6]=

0.1875

As expected, it is a rough estimate (the expected value is

θ=1/5=0.2

), since a small value was chosen for

. If one increases

, with a higher probability, one can get a better estimate of the phase.

Estimate the phase (the expected value is 1/5=0.2) for

=6:

In[7]:=

With[{m=N@QuantumCircuitOperator[{"PhaseEstimation",QuantumOperator[{"Phase",2Piphase}],6}][]},FromDigits[Keys[TakeLargest[m["Probabilities"],1]]〚1〛["Name"],2]/2^6//N]

Out[7]=

0.203125

External Links

Wolfram/QuantumFramework

Publisher Information

Contributed by: Wolfram Research, Quantum Computation Framework Team