A Resource-Efficient 4-Qubit Circuit for Bit-Flip Error Correction Using Feynman Gates

Dimitrios Gryllakis; Kyriakos N. Sgarbas

doi:10.2478/qic-2025-0019

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A Resource-Efficient 4-Qubit Circuit for Bit-Flip Error Correction Using Feynman Gates

Quantum Information & Computation

Volume 25 (2025): Issue 4 (August 2025)

By: Dimitrios Gryllakis and Kyriakos N. Sgarbas

Open Access

|Aug 2025

Figures & Tables

Gate-Level Diagram—Part 1 of the Proposed Error Correction Circuit (Correction of q[2] and q[1]).

Gate-Level Diagram—Part 2 of the Proposed Circuit (Verification and Adjustment of q[0]).

Gate-Level Diagram—Part 3 of the Proposed Circuit (Final Correction Based on Auxiliary State).

Repeated Subcircuit Block (Used in All Three Parts of the Circuit).

Block Diagram of the Full 4-Qubit Bit-Flip Error Correction Architecture.

Matrix representation of a 3-qubit layout result_

1	1	1	0	1	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	1	0	1	1	1

Substructure-Block Truth Table_

Column1	Column2	Column3
Row	Input State	Output State
R1	0000	1000
R2	0001	0001
R3	0010	0010
R4	0011	0011
R5	0100	0100
R6	0101	0101
R7	0110	0110
R8	0111	1111
R9	1000	0000
R10	1001	1001
R11	1010	1010
R12	1011	1011
R13	1100	1100
R14	1101	1101
R15	1110	1110
R16	1111	0111

Substructure-Block Unitary Matrix_

0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1
1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0

Simulation results showing fidelity comparison across all test cases_

Error State	Target State	Detection Accuracy	Correction Fidelity
\|001〉	\|000〉	100%	100%
\|010〉	\|000〉	100%	100%
\|100〉	\|000〉	100%	100%
\|110〉	\|111〉	100%	100%
\|101〉	\|111〉	100%	100%
\|011〉	\|111〉	100%	100%
\|111〉	\|111〉	100%	100%
\|000〉	\|000〉	100%	100%

Matrix representation of a 3-qubit layout process_

(000\|000)	(000\|000)	(000\|000)	(000\|111)	(000\|000)	(000\|111)	(000\|111)	(000\|111)
(001\|000)	(001\|000)	(001\|000)	(001\|111)	(001\|000)	(001\|111)	(001\|111)	(001\|111)
(010\|000)	(010\|000)	(010\|000)	(010\|111)	(010\|000)	(010\|111)	(010\|111)	(010\|111)
(011\|000)	(011\|000)	(011\|000)	(011\|111)	(011\|000)	(011\|111)	(011\|111)	(011\|111)
(100\|000)	(100\|000)	(100\|000)	(100\|111)	(100\|000)	(100\|111)	(100\|111)	(100\|111)
(101\|000)	(101\|000)	(101\|000)	(101\|111)	(101\|000)	(101\|111)	(101\|111)	(101\|111)
(110\|000)	(110\|000)	(110\|000)	(110\|111)	(110\|000)	(110\|111)	(110\|111)	(110\|111)
(111\|000)	(111\|000)	(111\|000)	(111\|111)	(111\|000)	(111\|111)	(111\|111)	(111\|111)

Comparison of Error-Correcting Circuit Approaches_

Feature	Proposed Circuit	Shor Code (9-Qubit)	Steane Code (7-Qubit)
Total Qubits Required	4	9	7
Correctable Error Types	Bit-flip	Bit + Phase	Bit + Phase
Ancilla Qubits Used	1	≥3	≥2
Gate Types Used	CNOT, NOT	CNOT, H, S, T	CNOT, H, S, T
Circuit Depth (approx.)	Low	High	High
Implementation Complexity	Low	High	High
Detection Accuracy (simulated)	100%	100% (theoretical)	100% (theoretical)
Fidelity (ideal simulation)	100%	100%	100%
Simulation Tool Used	IBM Composer	Theoretical	Theoretical

Truth table of the intended operation_

Column1	Column2	Column3
	Input State	Expected Output State
0	\|000>	\|000>
1	\|001>	\|000>
2	\|010>	\|000>
3	\|011>	\|111>
4	\|100>	\|000>
5	\|101>	\|111>
6	\|110>	\|111>
7	\|111>	\|111>

References

Authors

Metrics

Articles in this issue

DOI: https://doi.org/10.2478/qic-2025-0019 | Journal eISSN: 3106-0544 | Journal ISSN: 1533-7146

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Language: English

Page range: 344 - 355

Submitted on: May 13, 2025

Accepted on: Jun 27, 2025

Published on: Aug 22, 2025

Published by: Cerebration Science Publishing Co., Limited

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

ancilla qubit,

controlled-NOT gate,

quantum error detection,

quantum fault tolerance,

bit-flip error,

quantum circuit design

Related subjects:

Physics,

Computer sciences in the humanities,

Computer sciences,

Computer sciences in natural sciences

© 2025 Dimitrios Gryllakis, Kyriakos N. Sgarbas, published by Cerebration Science Publishing Co., Limited
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 25 (2025): Issue 4 (August 2025)

(000\|000)	(000\|000)	(000\|000)	(000\|111)	(000\|000)	(000\|111)	(000\|111)	(000\|111)
(001\|000)	(001\|000)	(001\|000)	(001\|111)	(001\|000)	(001\|111)	(001\|111)	(001\|111)
(010\|000)	(010\|000)	(010\|000)	(010\|111)	(010\|000)	(010\|111)	(010\|111)	(010\|111)
(011\|000)	(011\|000)	(011\|000)	(011\|111)	(011\|000)	(011\|111)	(011\|111)	(011\|111)
(100\|000)	(100\|000)	(100\|000)	(100\|111)	(100\|000)	(100\|111)	(100\|111)	(100\|111)
(101\|000)	(101\|000)	(101\|000)	(101\|111)	(101\|000)	(101\|111)	(101\|111)	(101\|111)
(110\|000)	(110\|000)	(110\|000)	(110\|111)	(110\|000)	(110\|111)	(110\|111)	(110\|111)
(111\|000)	(111\|000)	(111\|000)	(111\|111)	(111\|000)	(111\|111)	(111\|111)	(111\|111)

A Resource-Efficient 4-Qubit Circuit for Bit-Flip Error Correction Using Feynman Gates

Figures & Tables

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Matrix representation of a 3-qubit layout result_

Substructure-Block Truth Table_

Substructure-Block Unitary Matrix_

Simulation results showing fidelity comparison across all test cases_

Matrix representation of a 3-qubit layout process_

Comparison of Error-Correcting Circuit Approaches_

Truth table of the intended operation_

Paradigm

My account

1	1	1	0	1	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	1	0	1	1	1

0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1
1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0

1	1	1	0	1	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	1	0	1	1	1

0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1
1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0

1	1	1	0	1	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0
0	0	0	1	0	1	1	1

0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1
1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0
0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0