unitaryGate

Unitary matrix gate

Since R2023b

Installation Required: This functionality requires MATLAB Support Package for Quantum Computing.

Syntax

cg = unitaryGate(targetQubits,U)

cg = unitaryGate(targetQubits,U,RotationThreshold=thresh)

Description

cg = unitaryGate(targetQubits,U) returns a quantum.gate.CompositeGate object that applies a unitary matrix to the target qubits up to a global phase, that is, scaled by a constant factor.

cg = unitaryGate(targetQubits,U,RotationThreshold=thresh) also removes single-qubit rotation gates that have an angle magnitude less than the rotation threshold.

Examples

Unitary Matrix Gate and Its Matrix Representation

Create a unitary matrix gate that applies a unitary matrix to a single qubit.

U = (1/sqrt(3))*[1 -1+1i; 1+1i 1]

U = 2×2 complex

   0.5774 + 0.0000i  -0.5774 + 0.5774i
   0.5774 + 0.5774i   0.5774 + 0.0000i

cg = unitaryGate(1,U)

cg = 
  CompositeGate with properties:

             Name: ""
    ControlQubits: [1×0 double]
     TargetQubits: 1
            Gates: [3×1 quantum.gate.SimpleGate]

Get the matrix representation of the gate. Here, M is equal to U because the global phase is 1.

M = getMatrix(cg)

M = 2×2 complex

   0.5774 + 0.0000i  -0.5774 + 0.5774i
   0.5774 + 0.5774i   0.5774 + 0.0000i

Plot the returned unitary matrix gate to show its internal gates.

plot(cg)

Unitary Matrix Gate Acting on Two Qubits

Create a unitary matrix gate that applies a unitary matrix to two qubits with indices 1 and 2.

U = [1 0 0 0; 0 1 0 0; 0 0 1i 0; 0 0 0 1i];
cg = unitaryGate(1:2,U)

cg = 
  CompositeGate with properties:

             Name: ""
    ControlQubits: [1×0 double]
     TargetQubits: [1 2]
            Gates: [12×1 quantum.gate.QuantumGate]

Plot the unitary matrix gate. The unitaryGate function uses an algorithm that can represent any unitary matrix to return a unitary matrix gate, so it may not be a minimal representation of the input matrix.

plot(cg)

Unitary Matrix Gate Acting on Multiple Qubits

Create a unitary matrix gate that applies a random unitary matrix to three qubits with indices 1, 2, and 3.

U = orth(rand(2^3,"like",1j));
cg = unitaryGate(1:3,U)

cg = 
  CompositeGate with properties:

             Name: "unitary"
    ControlQubits: [1×0 double]
     TargetQubits: [1 2 3]
            Gates: [7×1 quantum.gate.CompositeGate]

Plot the unitary matrix gate.

plot(cg)

Calculate the global phase.

M = getMatrix(cg);
globalPhase = U(:)'*M(:)/norm(M,'fro')/norm(U,'fro')

globalPhase = -0.9762 - 0.2167i

Verify that the norm of M - globalPhase*U is 0, within machine precision.

norm(M - globalPhase*U)

ans = 3.1653e-15

Input Arguments

targetQubits — Target qubits of gate

positive integer scalar | positive integer vector

Target qubits of the gate, specified as a positive integer scalar index or vector of qubit indices.

Example: 1

Example: 3:5

U — Unitary matrix

square matrix

Unitary matrix, specified as a square matrix. The matrix must be of size $$2^n$$, where n is the number of target qubits.

thresh — Rotation threshold

2*pi*eps (default) | positive real number | "auto" | "none"

Rotation threshold, specified as one of these values:

• positive real number

• "auto" — Set the threshold to the default value of 2*pi*eps.

• "none" — Do not remove gates.

The unitaryGate function removes single-qubit rotation gates that have an angle magnitude less than this rotation threshold.

More About

Unitary Matrix

An invertible complex square matrix U is unitary if its conjugate transpose is also its inverse, that is, if U*U = UU* = I.

Tips

• You can use the unitaryGate function to decompose any unitary matrix applied to n target qubits into a composite gate of O(4ⁿ) simple quantum gates. However, the resulting CompositeGate object may not contain the minimal number of gates for an input matrix.

References

[1] Shende, Vivek V., Stephen S. Bullock, and Igor L. Markov. "Synthesis of Quantum Logic Circuits." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, no. 6 (June 2006): 1000–1010. https://doi.org/10.1109/TCAD.2005.855930.

[2] Vatan, Farrokh, and Colin Williams. "Optimal Quantum Circuits for General Two-Qubit Gates." Physical Review A 69, no. 3 (March 22, 2004): 032315. https://doi.org/10.1103/PhysRevA.69.032315.

[3] Drury, Byron, and Peter J. Love. "Constructive Quantum Shannon Decomposition from Cartan Involutions." Journal of Physics A: Mathematical and Theoretical 41, no. 39 (October 3, 2008): 395305. https://doi.org/10.1088/1751-8113/41/39/395305.

Version History

Introduced in R2023b