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Quantum Computation

Harrow–Hassidim–Lloyd (HHL) Algorithm

Numerically solve a system of linear equations using a quantum algorithm

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The HHL algorithm is a quantum algorithm designed to solve systems of linear equations. Given a matrix A and a vector b, it tries to solve

A·x=b

Install the QuantumFramework paclet:

In[1]:=

PacletInstall["Wolfram/QuantumFramework"];

Load the paclet:

In[2]:=

Needs["Wolfram`QuantumFramework`"];

Consider a Hermitian operator of 2-qubits, a random vector

and 6 ancillary qubits:

In[3]:=

prec=6;A={{0.21`,0.15`,0.07`,0.`},{0.15`,0.38`,0.18`,0.07`},{0.07`,0.18`,0.19`,0.04`},{0.`,0.07`,0.04`,0.22`}};b=Round[RandomReal[1,2^2],0.01];

Generate an HHL circuit that takes

and number of ancillary qubits

prec

In[4]:=

hhl=Block[{d=Log2@Length[b],qpe},QuantumCircuitOperator[{QuantumCircuitOperator[{QuantumState[Normalize[b]]1+prec+Range[d]},"State preparation"],qpe=QuantumCircuitOperator["PhaseEstimation"[QuantumOperator[MatrixExp[2PiIA]],prec],1+Range[prec+d]]〚d+1;;-prec-1〛,QuantumCircuitOperator["GrayOracle"[If[#0,0,1./#]&,prec,1],"Eigenvalue Inversion"],qpe["Dagger"],QuantumOperator[{QuantumState["Register"[prec+1,2^prec]]["Dagger"]Range[prec+1]},"Label""Post-selection"]}]];

Show the HHL circuit diagram:

In[5]:=

hhl"Diagram","ShowGateLabels"False,



Out[5]=

Given the result of the above circuit, find the corresponding probabilities, and then their square roots to get the solution with the HHL solver:

In[6]:=

Sqrt@hhl[]["ProbabilitiesList"]

Out[6]=

{0.642053,0.72234,0.176279,0.186864}

Although the probabilities can be measured experimentally, they do not all have the correct signs.

Find the corresponding amplitudes, and then normalize them to get the solution with the HHL solver:

In[7]:=

Normalize@Values@hhl[]["Amplitudes"]//Chop

Out[7]=

{0.642053,0.72234,0.176279,-0.186864}

Find the corresponding solution using the classical approach:

In[8]:=

Normalize@LinearSolve[A,b]

Out[8]=

{0.641359,0.722972,0.172766,-0.190057}

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Contributed by: Wolfram Research, Quantum Computation Framework Team

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Harrow–Hassidim–Lloyd (HHL) Algorithm

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