Grover's Search Algorithm

Construct Grover's circuit to find solutions of a Boolean function

Grover's search algorithm is a quantum algorithm that efficiently finds a marked item in an unsorted database. The algorithm is also applicable to solving the satisfiability problem (SAT), where the goal is to determine if there exists an assignment of variables that satisfies a given Boolean formula.

The action of a Boolean oracle is defined by this transformation:

|x〉|q〉->|x〉|q⊕f(x)〉

with

|x〉

the index register state of n-qubits, and

|q〉

the state of an ancillary qubit carrying the result of Boolean function

f(x)

; meaning if

|q〉=|0〉

then it will be

|1〉

if x is a solution of f(x), and no change if not.

Define a Boolean function of 3-SAT with 5 clauses:

In[1]:=

bf=And[!v1||!v2||!v3,v1||!v2||v3,v1||v2||!v3,v1||!v2||!v3,!v1||v2||v3];

Find solutions such that bf is True:

In[2]:=

SatisfiabilityInstances[bf,{v1,v2,v3},All]

Out[2]=

{{True,True,False},{True,False,True},{False,False,False}}

Install and load the QuantumFramework paclet:

In[3]:=

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

Get the corresponding oracle's quantum circuit:

In[4]:=

oracle=QuantumCircuitOperator[{"BooleanOracle",bf}];

The diagram of the oracle:

In[5]:=

oracle["Diagram"]

Out[5]=

We shall prepare the 4-qubit in the above circuit (i.e., the ancillary qubit) in the 0 state, and then other qubits (1-3) in the index register states. To compare with the above truth table, we will create the index register states

|x〉

in the order

-1〉

down to

|0〉

In[6]:=

states=QuantumTensorProduct[#,QuantumState["0"]]&/@Table[QuantumState[{"Register",3,i}],{i,2^3-1,0,-1}];

Create a list with elements {|x〉,|q⊕f(x)}:

In[7]:=

TableForm[Transpose[{Table[QuantumState[{"Register",3,i}]["Formula"],{i,2^3-1,0,-1}],QuantumPartialTrace[oracle[#],{1,2,3}]["Formula"]&/@states}],TableHeadings{None,{"|x>","|q⊕f(x)>"}}]

Out[7]//TableForm=

\|x>	\|q⊕f(x)>
\|111〉	\|0〉
\|110〉	\|1〉
\|101〉	\|1〉
\|100〉	\|0〉
\|011〉	\|0〉
\|010〉	\|0〉
\|001〉	\|0〉
\|000〉	\|1〉

As one can see, the oracle circuit reproduces the same result as the classical ones.

The action of a phase oracle can be defined as the following transformation:

|x〉->

f(x)

(-1)

|x〉

with

|x〉

the index register state of n-qubits.

The corresponding oracle's quantum circuit:

In[8]:=

phaseOracle=QuantumCircuitOperator[{"PhaseOracle",bf}];

The diagram of the oracle:

In[9]:=

phaseOracle["Diagram"]

Out[9]=

We shall prepare the 3-qubit in the index register states. To compare with the above truth table, we will create the index register states

|0〉

in the order

-1〉

down to

|0〉

In[10]:=

states=Table[QuantumState[{"Register",3,i}],{i,2^3-1,0,-1}];

Create a list with elements

x>,

f(x)

(-1)

|x>}:

In[11]:=

TableFormTranspose[{Table[QuantumState[{"Register",3,i}]["Formula"],{i,2^3-1,0,-1}],phaseOracle[#]["Formula"]&/@states}],TableHeadingsNone,"|x>","-1

f(x)

)

|x>"

Out[11]//TableForm=

\|x>	-1 f(x) ) \|x>
\|111〉	\|111〉
\|110〉	-\|110〉
\|101〉	-\|101〉
\|100〉	\|100〉
\|011〉	\|011〉
\|010〉	\|010〉
\|001〉	\|001〉
\|000〉	-\|000〉

As one can see, the oracle circuit reproduces the same result as the classical ones.

Let's define a Boolean function with only one solution (note in our framework, it does not matter if there is one or more solutions):

In[12]:=

bf=And[q1,q2,q3];

Generate the corresponding Grover circuit using a phase oracle:

In[13]:=

QuantumCircuitOperator[{"GroverPhase",bf}]["Diagram"]

Out[13]=

Generate the corresponding Grover circuit using a Boolean oracle:

In[14]:=

QuantumCircuitOperator[{"Grover",bf}]["Diagram"]

Out[14]=

Consider the following Boolean function and explore different cases and how the probability of success of the algorithm depends on circuit design or the initial state of the ancillary qubit:

In[15]:=

g=QuantumCircuitOperator[{"GroverPhase",bf}];g["Diagram"]

Out[15]=

Initial state for the circuit (qubit 1-6, each one in |x+>):

In[16]:=

ψ0=QuantumState[{"UniformSuperposition",3}];

Apply the Grover operator 20 times:

In[17]:=

steps=NestList[g,ψ0,20];

Calculate the success probability after each iteration:

In[18]:=

success=QuantumState[{"Register",3,2^3-1}]["Dagger"][#]["Norm"]&/@steps;

Plot the success probability:

In[19]:=

ListLinePlot[success,GridLinesAutomatic,AxesLabel{"# of Grover iteration","Probability of success"},AspectRatio1/2]

Out[19]=

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Contributed by: Wolfram Research, Quantum Computation Framework Team

Wolfram Language Example Repository

Grover's Search Algorithm

See Also

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