Compute the Product of a Matrix and Vector using a Quantum Circuit

In the realm of quantum linear solvers, a typical procedure involves designing a circuit that yields the equivalent outcome of applying an

n×n

matrix, like

, to an

-element vector, such as

. This entails computing the operation

A.x

with a quantum circuit.

Install the QuantumFramework paclet:

In[1]:=

PacletInstall["Wolfram/QuantumFramework"];

Load the paclet:

In[2]:=

Needs["Wolfram`QuantumFramework`"];

Given n-qubits, the dimension of A will be

. Create a random operator for the 3-qubit case:

In[3]:=

n=3;a=RandomReal[{-1,1},{2^n,2^n}];A=QuantumOperator[a];

Find the Pauli decomposition of A:

In[4]:=

pauliDecompose=A["PauliDecompose"]

Out[4]=

XXX0.211297+0.,XXY0.-0.110715,XXZ0.198033+0.,XXI-0.0803537+0.,XYX0.-0.0877234,XYY-0.107238+0.,XYZ0.+0.0203499,XYI0.+0.0981309,XZX0.0598082+0.,XZY0.-0.142928,XZZ-0.171679+0.,XZI-0.151295+0.,XIX-0.258682+0.,XIY0.+0.0114687,XIZ-0.22355+0.,XII0.0388+0.,YXX0.+0.320737,YXY-0.0448872+0.,YXZ0.+0.412652,YXI0.-0.0655296,YYX-0.200803+0.,YYY0.+0.384275,YYZ-0.0289906+0.,YYI0.0256568+0.,YZX0.+0.341995,YZY-0.114837+0.,YZZ0.-0.0336461,YZI0.-0.0385934,YIX0.+0.138818,YIY0.199916+0.,YIZ0.+0.499497,YII0.-0.0537861,ZXX-0.325381+0.,ZXY0.+0.0876408,ZXZ0.140522+0.,ZXI0.0942942+0.,ZYX0.-0.209835,ZYY0.0881496+0.,ZYZ0.-0.105072,ZYI0.-0.153355,ZZX0.108864+0.,ZZY0.-0.0381291,ZZZ-0.119564+0.,ZZI-0.0789658+0.,ZIX-0.198655+0.,ZIY0.+0.201184,ZIZ0.0572329+0.,ZII-0.0628151+0.,IXX-0.20364+0.,IXY0.-0.119644,IXZ-0.0148875+0.,IXI-0.263927+0.,IYX0.+0.0190936,IYY0.14018+0.,IYZ0.-0.656842,IYI0.+0.0806824,IZX0.188248+0.,IZY0.+0.0737358,IZZ0.161554+0.,IZI-0.42743+0.,IIX0.211465+0.,IIY0.+0.12888,IIZ0.0794086+0.,III-0.136596+0.

Find the binary length of terms in the Pauli decomposition:

In[5]:=

L=Log2@Length[pauliDecompose]

Out[5]=

Create a multiplexer, using the keys of the Pauli decomposition:

In[6]:=

mp=QuantumCircuitOperator["Multiplexer"[Sequence@@Keys[pauliDecompose]]];

Show the circuit diagram:

In[7]:=

mp["Diagram","ShowGateLabels"False]

Out[7]=

Create a quantum state using the square root of values in the Pauli decomposition:

In[8]:=

ancillary=QuantumState[Sqrt@Values[pauliDecompose]]

Out[8]=

QuantumState

Pure state	Qudits: 6
Type: Vector	Dimension: 64
Picture: Schrödinger



Create a random n-element vector:

In[9]:=

x=RandomReal[{-1,1},2^n]

Out[9]=

{-0.612068,-0.368019,0.465311,0.960787,-0.320024,-0.0583026,-0.397421,-0.189865}

Create a circuit for implementing

A.x

In[10]:=

qc=N@QuantumCircuitOperatorancillary,QuantumState[x]Range[L+1,L+1+n],mp,

†

(ancillary["Conjugate"])

;

In[11]:=

qc["Diagram"]

Out[11]=

Calculate the amplitudes after running the circuit:

In[12]:=

qc[]["AmplitudesList"]

Out[12]=

{-1.52781+0.,0.127506+0.,-0.512799+0.,0.804974+0.,1.01403+0.,-0.398952+0.,0.590336+0.,-0.306825+0.}

Calculate

A.x

directly:

In[13]:=

a.x

Out[13]=

{-1.52781,0.127506,-0.512799,0.804974,1.01403,-0.398952,0.590336,-0.306825}

Check the results are the same:

In[14]:=

%-%%//Chop

Out[14]=

{0,0,0,0,0,0,0,0}

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Contributed by: Wolfram Research, Quantum Computation Framework Team

Wolfram Language Example Repository

Compute the Product of a Matrix and Vector using a Quantum Circuit

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