ucrzGate

Uniformly controlled z-axis rotation gate

Since R2023b

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

Syntax

cg = ucrzGate(controlQubits,targetQubit,theta)

cg = ucrzGate(controlQubits,targetQubit,theta,RotationThreshold=thresh)

Description

cg = ucrzGate(controlQubits,targetQubit,theta) applies a z-axis rotation gate of angle theta to a single target qubit for each computational basis state of the control qubits.

cg = ucrzGate(controlQubits,targetQubit,theta,RotationThreshold=thresh) also removes single-qubit rotation gates that have an angle magnitude less than the rotation threshold.

Examples

Uniformly Controlled z-Axis Rotation Gate and Its Matrix Representation

Create a uniformly controlled z-axis rotation gate that acts on a control qubit with index 1 and a target qubit with index 2 using rotation angles pi/2 and pi.

cg = ucrzGate(1,2,[pi/2 pi])

cg = 
  CompositeGate with properties:

             Name: "ucrz"
    ControlQubits: [1x0 double]
     TargetQubits: [1 2]
            Gates: [4x1 quantum.gate.SimpleGate]

Get the matrix representation of the gate.

M = getMatrix(cg)

M = 4×4 complex

   0.7071 - 0.7071i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.7071 + 0.7071i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 - 1.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 1.0000i

Plot the returned UCRZ gate to show its internal gates.

plot(cg)

Uniformly Controlled z-Axis Rotation Gate Using Rotation Threshold

Create a uniformly controlled z-axis rotation gate that acts on three control qubits with indices 1, 2, and 3 and a target qubit with index 4 using eight rotation angles.

theta = rand(8,1);
cg1 = ucrzGate(1:3,4,theta)

cg1 = 
  CompositeGate with properties:

             Name: "ucrz"
    ControlQubits: [1x0 double]
     TargetQubits: [1 2 3 4]
            Gates: [16x1 quantum.gate.SimpleGate]

Plot the UCRZ gate.

plot(cg1)

Now create the same gate, but specify a rotation threshold of 0.1.

cg2 = ucrzGate(1:3,4,theta,RotationThreshold=0.1)

cg2 = 
  CompositeGate with properties:

             Name: "ucrz"
    ControlQubits: [1x0 double]
     TargetQubits: [1 2 3 4]
            Gates: [10x1 quantum.gate.SimpleGate]

Plot the UCRZ gate. The specified rotation threshold removes multiple gates from the final composite gate.

plot(cg2)

Input Arguments

controlQubits — Control qubits of gate

positive integer scalar | positive integer vector

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

Example: 2

Example: 6:8

targetQubit — Target qubit of gate

positive integer scalar

Target qubit of the gate, specified as a positive integer scalar index.

Example: 1

theta — Rotation angles

real scalar | real vector

Rotation angles, specified as a real scalar if the number of control qubits is 0 or real vector of length $$2^n$$, where n is the number of control qubits.

Example: pi

Example: (1:4)*pi/2

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

Output Arguments

cg — UCRZ gate

CompositeGate object | SimpleGate object

UCRZ gate, returned as a CompositeGate object or a SimpleGate object.

• If the number of control qubits is greater than 0, then the ucrzGate function returns cg as a CompositeGate object that consists of several internal gates.

• If no control qubits are specified, then the ucrzGate function returns cg as the RZ gate (rzGate).

More About

Matrix Representation of Uniformly Controlled z-Axis Rotation Gate

The matrix representation of a uniformly controlled z-axis rotation gate has a block diagonal structure made from the 2-by-2 RZ gate matrices of the input rotation angles. For example, when the UCRZ gate is applied to control qubit 1 and target qubit 2 with a rotation angle vector of $$\left[\begin{array}{cc}{\theta }_1& {\theta }_2\end{array}\right]$$, the matrix representation is

$$\left[\begin{array}{cccc}\exp \left(-i\frac{{\theta }_1}{2}\right)& 0& 0& 0\\ 0& \exp \left(i\frac{{\theta }_1}{2}\right)& 0& 0\\ 0& 0& \exp \left(-i\frac{{\theta }_2}{2}\right)& 0\\ 0& 0& 0& \exp \left(i\frac{{\theta }_2}{2}\right)\end{array}\right].$$

References

[1] Möttönen, Mikko, Juha J. Vartiainen, Ville Bergholm, and Martti M. Salomaa. "Quantum Circuits for General Multiqubit Gates." Physical Review Letters 93, no. 13 (September 20, 2004): 130502. https://doi.org/10.1103/PhysRevLett.93.130502.

[2] Camps, Daan, and Roel Van Beeumen. "FABLE: Fast Approximate Quantum Circuits for Block-Encodings." In 2022 IEEE International Conference on Quantum Computing and Engineering (QCE), 104–113. Broomfield, CO, USA: IEEE, 2022. https://doi.org/10.1109/QCE53715.2022.00029.

Version History

Introduced in R2023b