Nov 28, 2024

Public workspaceBSA: The Bloch Sphere Approach as a Geometrical Design Tool for Building Cost-Effective Quantum Gates V.4

DOI
https://dx.doi.org/10.17504/protocols.io.bp2l6dkkdvqe/v4
BSA: The Bloch Sphere Approach as a Geometrical Design Tool for Building Cost-Effective Quantum Gates
  • Ali Al-Bayaty1,
  • Marek Perkowski1
  • 1Portland State University
  • Ali Al-Bayaty: Department of Electrical and Computer Engineering
  • Marek Perkowski: Department of Electrical and Computer Engineering
Ali Al-Bayaty
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DOI: https://dx.doi.org/10.17504/protocols.io.bp2l6dkkdvqe/v4
Protocol Citation: Ali Al-Bayaty, Marek Perkowski 2024. BSA: The Bloch Sphere Approach as a Geometrical Design Tool for Building Cost-Effective Quantum Gates. protocols.io https://dx.doi.org/10.17504/protocols.io.bp2l6dkkdvqe/v4Version created by Ali Al-Bayaty
License: This is an open access protocol distributed under the terms of the Creative Commons Attribution License,  which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited
Protocol status: Working
We use this protocol and it's working
Created: November 28, 2024
Last Modified: November 28, 2024
Protocol Integer ID: 113103
Keywords: Bloch sphere, XY-plane, quantum gates, quantum rotations, geometrical visualization tool, geometrical verification tool, geometrical design tool, cost-effective gates, generic gates, Boolean, Phase, quantum layouts, quantum computer, Bloch sphere approach, quantum circuits, Clifford+T, effective quantum gates the bloch sphere, bloch sphere, bloch sphere approach, axis of the bloch sphere, plane of the bloch sphere, open geometrical framework for prospective quantum computing research, native gates for such quantum computer, other projectional planes of the bloch sphere, quantum gate, quantum operations of all ibm, effective quantum gates for other quantum computer, ibm quantum computer, innovative quantum gate, quantum computing, other quantum computer, effective quantum gate, rotational quantum operation, prospective quantum computing research, such quantum computer, series of quantum gate, quantum library, qubit, rotational quantum operations in the xy, quantum operation, geometrical 3d sphere, part of the ibm qiskit ecos
Disclaimer
The Bloch sphere approach (BSA) belongs to our manuscripts that are licensed under the CC BY-NC-ND 4.0 International License, which permits any non-commercial use, sharing, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if you modified the licensed material. You do not have permission under these licenses to share adapted material derived from our manuscripts or parts of them. The methods, images, or other third-party material in our manuscripts are included in the manuscript’s Creative Commons license, unless indicated otherwise in a credit line to the material. To view a copy of these licenses, visit http://creativecommons.org/licenses/by-nc-nd/4.0/
Abstract
The Bloch sphere is a geometrical 3D sphere that visualizes the states of a qubit after a series of quantum gates are applied to it. In quantum computing, the Bloch sphere is mainly used as a geometrical visualization (and verification) tool.

On the other hand, in this protocol, we introduce the Bloch sphere as a geometrical design tool for building cost-effective quantum gates based on their rotational quantum operations in the XY, XZ, and/or YZ planes, which are the 2D circular projections along the axes of the Bloch sphere. Collectively, the Bloch sphere and its projectional planes are termed the Bloch sphere approach (BSA).

With the BSA, various generic and cost-effective quantum gates and libraries are designed for IBM quantum computers, using the symmetrical and semi-symmetrical structures [1-5], Clifford+T gates, and IBM native "basis" gates ( , , , and ). Our designed generic and cost-effective quantum gates and libraries are listed as follows, where 2 n 5 qubits.

  1. Quantum libraries: GALA-n [3, 6] and CALA-n [4, 7], which have become part of the IBM Qiskit ecosystem [8]
  2. n-bit Toffoli gate [2-4]
  3. n-bit Boolean gates (AND, NAND, OR, NOR, Implication, and Inhibition) [3, 4]
  4. n-bit controlled- ( ) and controlled- ( ) gates [3, 4]
  5. n-bit Fredkin gate [3, 4]
  6. n-bit Miller gate [3, 4]
  7. Boolean-Phase SWAP gate: p-SWAP [4, 5]

Because the quantum operations of all IBM native gates mainly rotate around the X-axis and Z-axis of the Bloch sphere, the XY-plane of the Bloch sphere is utilized here for the BSA. However, the BSA can also be utilized to build generic and cost-effective quantum gates for other quantum computers, e.g., Intel, Google, and Rigetti, using other projectional planes of the Bloch sphere, e.g., the XZ and YZ planes, based on the supported native gates for such quantum computers. Therefore, in this protocol, we introduce the BSA as a generic and open geometrical framework for prospective quantum computing research and building interesting and innovative quantum gates and circuits.
Troubleshooting
Preliminary Notes

Note

The Bloch sphere consists of three axes (X-axis in blue, Y-axis in red, and Z-axis in green) with their corresponding rotational angles (θX, θY, and θZ), where "+θ" denotes a counterclockwise rotational angle and "−θ" denotes a clockwise rotational angle, in radians.



Note
  1. The Clifford+T gates are .
  2. For the BSA, based on IBM native gates, we limit the Clifford+T gates to , which is denoted by CTG0.
  3. The gates of CTG0 rotating around the X-axis of the Bloch sphere are .
  4. The gates of CTG0 rotating around the Z-axis of the Bloch sphere are .
  5. The gate of CTG0 is primarily used in the BSA, to transform the state of a qubit from the Z-axis ( or ) to the XY-plane ( or ), respectively, and vice versa.
  6. In general, the , , and can be used as superposition gates. However, in the BSA, different geometrical analyses should be taken into account when using and as superposition gates to build cost-effective gates for various quantum computers.
  7. For n 2 qubits, an n-bit quantum gate has n−1 controls (input qubits) and one target (output qubit), except for the SWAP and Fredkin (CSWAP) gates.
  8. In the BSA, all controls are connected to the target using gates, and the target has a defined number of CTG0 gates . Every gate flips the target's state in the perimeter of the XY-plane by radians, i.e., around the X-axis of the Bloch sphere, when its control is in the state.
  9. When all controls are connected to the target using gates, this allows to map the target among the controls, and an n-bit quantum gate is then cost-effectively fitted into the layout (architecture) of any quantum computer, e.g., IBM, Intel, Google, and Rigetti. Hence, SWAP gates are never utilized to connect the non-neighboring physical qubits of a quantum computer.


Note
The BSA geometrically builds cost-effective n-bit quantum gates of symmetrical and semi-symmetrical structures, by assigning appropriate angles to their rotational gates using the projectional planes of the Bloch sphere, where n 2. The following figures demonstrate arbitrary n-bit symmetrical quantum gates, where:
  • SP1 and SP2 are the superposition gates: , , and/or .
  • θ's are the Clifford+T gates and/or the rotational angles of IBM native RZ( θ) gates. Note that θ's can have similar, semi-similar, or non-similar values (as rotational quantum gates).

The 2-bit symmetrical quantum gate.


The 3-bit symmetrical quantum gate.


The 4-bit symmetrical quantum gate.


The 4-bit symmetrical quantum gate.



Note
The XY-plane of the Bloch sphere is divided into segments to represent the rotational quantum operations of Clifford+T gates and IBM native gates around the Z-axis as follows, where the black dots represent these segments as shown below.
  1. The semicircle segment is half of the XY-plane, to represent the quantum rotations of gates. Such that, the XY-plane has two semicircles.
  2. The quadrant segment is one-fourth of the XY-plane, to represent the quantum rotations of and gates. Such that, the XY-plane has four quadrants.
  3. The octant segment is one-eighth of the XY-plane, to represent the quantum rotations of and gates. Such that, the XY-plane has eight octants.
The XY-plane visualizes the rotational quantum operations of Cifford+T gates around the Z-axis of the Bloch sphere, when a qubit is initially set to the state.



Note
Similarly, the XZ-plane of the Bloch sphere is divided into segments (semicircles, quadrants, and octants) to represent the rotational quantum operations based on the native gates of a quantum computer, as illustrated below where the black dots represent the segments (semicircles, quadrants, and octants).
The XZ-plane visualizes the rotational quantum operations of native gates for a quantum computer around the Y-axis of the Bloch sphere, when a qubit is initially set to the state.



Note
Similarly, the YZ-plane of the Bloch sphere is divided into segments (semicircles, quadrants, and octants) to represent the rotational quantum operations based on the native gates of a quantum computer, as demonstrated below where the black dots represent the segments (semicircles, quadrants, and octants).
The YZ-plane visualizes the rotational quantum operations of native gates for a quantum computer around the X-axis of the Bloch sphere, when a qubit is initially set to the state.



Note
In general, the Bloch sphere approach (BSA) is a generic and open geometrical framework for building cost-effective quantum gates, based on the Clifford+T gates and the supported native gates of any quantum computer, e.g., IBM, Intel, Google, and Rigetti.

For a quantum computer, various projectional planes (P) of the Bloch sphere can be combined and employed, depending on the operational purpose of a quantum gate that needs to be geometrically designed. The following figures depict arbitrary P around the utilized axes of the Bloch sphere, where the black dots represent the segments (semicircles, quadrants, and octants).
Arbitrary m projectional planes (P) around the X-axis of the Bloch sphere, where m 1. Note the YZ-plane































Arbitrary m projectional planes (P) around the Y-axis of the Bloch sphere, where m 1. Note the XZ-plane































Arbitrary m projectional planes (P) around the Z-axis of the Bloch sphere, where m 1. Note the XY-plane

Arbitrary m projectional planes (P) around all three axes of the Bloch sphere, where m 1.


The BSA Protocol (for IBM quantum computers)
For symmetrical and semi-symmetrical structures [1-5], transform the target's state of an n-bit quantum gate from the Z-axis of the Bloch sphere into the XY-plane using one gate, where n 2 qubits. Note that the target is initially set to either or state, depending on the operational purpose of such a quantum gate.
For the target, define all Clifford+T gates (based on IBM native gates) as the set: CTG0 = .
For the target, define all segments of the XY-plane as the set: SEG0 = {semicircles, quadrants, octants}.
If the target is not controlling other qubits, then CTG0 is limited to the set: CTG1 = . Otherwise, CTG1 = CTG0.
Since SEG0 only identifies quantum gates rotating around the Z-axis of the Bloch sphere, then CTG1 is limited to the set: CTG2 = .
Note
Here, we assume that the 1-bit quantum rotations around the X-axis of the Bloch sphere are not required for the target. Otherwise, this step is negligible and CTG2 = CTG1, but this will affect the selection of quantum rotations in the next steps!

Let counts the total number of gates from the controls to the target, and the new sets of CTG and SEG will be defined as follows.
If , then CTG3 = , and SEG1 = {quadrants, octants}.
If , then a set of IBM native RZ gates defines CTG3 = {RZ(θ1), RZ(θ2), ...}, where θ's for arbitrary SEG1.
If , then CTG3 = , and SEG1 = {octants}.
If , then a set of IBM native RZ gates defines CTG3 = {RZ(θ1), RZ(θ2), ...}, where θ's for arbitrary SEG1.
The and permutative gates of CTG3 define the desirable quantum operation (logic and behavior) of an n-bit quantum gate, which can be designed and visualized based on the SEG1 of the XY-plane.
Re-transform the final reached target's state of an n-bit quantum gate from the XY-plane into the Z-axis of the Bloch sphere using another gate.
Finally, the re-transformed target's state is considered the output of such a quantum gate. For instance, the target in the state indicates a solution (True), while its state indicates a non-solution (False).
Protocol references
[1] A. Barenco, C.H. Bennett, R. Cleve, D.P DiVincenzo, N. Margolus, P. Shor, T. Sleator, J.A. Smolin, and H. Weinfurter, "Elementary gates for quantum computation," Physical Review A, vol. 52, no. 5, p. 3457, 1995.

[2] A. Al-Bayaty and M. Perkowski, "Cost-effective realization of n-bit Toffoli gates for IBM quantum computers using the Bloch sphere approach and IBM native gates," 2024, arXiv:2410.13104.

[3] A. Al-Bayaty and M. Perkowski, "GALA-n: Generic architecture of layout-aware n-bit quantum operators for cost-effective realization on IBM quantum computers," 2023, arXiv:2311.06760.

[4] A. Al-Bayaty, X. Song, and M. Perkowski, "CALA-n: A quantum library for realizing cost-effective 2-, 3-, 4-, and 5-bit gates on IBM quantum computers using Bloch sphere approach, Clifford+T gates, and layouts," 2024, arXiv:2408.01025.

[5] A. Al-Bayaty and M. Perkowski, "p-SWAP: A generic cost-effective quantum Boolean-phase SWAP gate using two CNOT gates and the Bloch sphere approach," 2024, arXiv:2410.16641.

[6] A. Al-Bayaty. "GALA-n Quantum Library." GITHUB.com. Accessed: Sep. 19, 2024. [Online.] Available: https://github.com/albayaty/gala_quantum_library

[7] A. Al-Bayaty. "CALA-n Quantum Library." GITHUB.com. Accessed: Sep. 19, 2024. [Online.] Available: https://github.com/albayaty/cala_quantum_library

[8] IBM Quantum Computing. "Qiskit Ecosystem." IBM.com. Accessed: Sep. 19, 2024. [Online.] Available: https://www.ibm.com/quantum/ecosystem