Optimal T-depth Quantum Circuits for Implementing Arbitrary Boolean Functions

Optimal T-depth quantum circuit synthesis for various standard S-boxes with corresponding resource estimates using Theorem 2. *A detailed analysis for the AES S-box is provided in Section 4.

S-box	#Variables	Ancilla count	CNOT count	CNOT depth	T-count	T-depth
LowMC [22]	3	9	39	13	12	1
DEFAULT [23]	4	17	79	27	24	2
GIFT [24]	4	14	76	27	24	2
PRESENT [25]	4	19	105	30	32	2
PRINCE [26]	4	24	128	28	40	2
ASCON [27]	5	27	120	22	32	1
AES [28]*	8	596	3647	184	984	3

Toffoli decomposition with measurement-based uncomputation, using four T gates, (a) without using any ancilla qubit [20], (b) with a single reusable ancilla qubit [13].

Table 2.

Quantum circuits for AES S-box with corresponding resource estimates.

Toffoli-to-T	Ancilla count	CNOT count	CNOT depth	T-count	T-depth
Using Figure 1a	470	2909	175	984	4
Using Figure 1b	596	3647	184	984	3

Quantum circuit implementing a 3-bit S-box (used in LowMC) with T-depth 1.

Quantum circuit implementing f (x₀, x₁, x₂, x₃) = x₀x₂ ⊕ x₁x₃ ⊕ x₀x₁x₂x₃ with T-depth 2.

Table 3.

Quantum implementation of a single round of AES with corresponding resource estimates.

Toffoli-to-T	Ancilla count	CNOT count	CNOT depth	T-count	T-depth
Using Figure 1a	7520	47064	189	15744	4
Using Figure 1b	9536	58872	198	15744	3

Table 4.

Optimal T-depth quantum implementation of full round AES with corresponding resources.

Key size	No. of round	Ancilla count	CNOT count	CNOT depth	T-count	T-depth
128	10	10688	588328	1967	157440	30
192	12	10944	706072	2363	188928	36
256	14	11200	823816	2759	220416	42

Table 5.

Optimal T-depth quantum implementation of full round AES: A comparison with prior results.

Key size	References	Ancilla count	CNOT count	CNOT depth	T-count	T-depth
128	[13, Table 4]	4244	284420	NA	54400	120
128	[30, Table 2]	9384	NA	NA	33600	50
128	[15, Table 13]	3689	132376	NA	27200	40
128	[14, Table 7]	5576	285393	NA	62400	30
128	[16, Table VI]	NA	228020	NA	52800	30
128	[18, Table 8]	NA	176580	NA	33600	30
128	[19, Table 5]	6128	120812	NA	117984	30
128	[This paper]	10688	588328	1967	157440	30
192	[13, Table 4]	4564	321021	NA	60928	144
192	[30, Table 2]	10456	NA	NA	37632	60
192	[15, Table 13]	3945	149256	NA	30464	48
192	[19, Table 5]	6448	136812	NA	132960	36
192	[This paper]	10944	706072	2363	188928	36
256	[13, Table 4]	4884	393534	NA	75072	168
256	[30, Table 2]	46368	NA	NA	12704	70
256	[15, Table 13]	4457	187128	NA	38080	56
256	[19, Table 5]	6768	168548	NA	165264	42
256	[This paper]	11200	823816	2759	220416	42

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Articles in this issue

DOI: https://doi.org/10.2478/qic-2025-0032 | Journal eISSN: 3106-0544 (formerly 1533-7146) | Journal ISSN: 1533-7146

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Language: English

Page range: 586 - 597

Submitted on: Jun 1, 2025

Accepted on: Sep 1, 2025

Published on: Mar 9, 2026

Published by: Cerebration Science Publishing Co., Limited

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

AES,

block cipher,

Boolean function,

cryptanalysis,

efficient circuit implementation,

S-box synthesis,

T-depth,

universal gate sets

Related subjects:

Computer sciences,

Computer sciences in the humanities,

Computer sciences in natural sciences,

Mathematics,

Applied mathematics,

Physics,

Quantum physics

© 2026 Suman Dutta, Anik Basu Bhaumik, Anupam Chattopadhyay, Subhamoy Maitra, 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 6 (December 2025)

Optimal T-depth Quantum Circuits for Implementing Arbitrary Boolean Functions

Figures & Tables

Table 1.

Figure 1.

Table 2.

Figure 2.

Figure 3.

Table 3.

Table 4.

Table 5.

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