Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.11153

Optimal T-Count for Block Encodings of Fermionic and Spin Hamiltonians

Jiaxin Ma, Kevin J. Joven, Yuan Liu

quant-phSubmitted Sep 10, 2026• Announced Sep 11, 2026• 0 citations
arXivHTMLPDF

Abstract

We determine the non-Clifford $T$-gate cost of constructing block encodings of structured fermionic and spin Hamiltonians in a unitary Clifford$+T$ model, when arbitrarily many clean ancillas and unrestricted block-encoding subnormalization are allowed, but without mid-circuit measurements or classical feed-forward. Our main technical tool is an ancilla-compression theorem: any block encoding of an $n$-qubit operator with $a$ clean ancillas and at most $s$ $T$ gates can be compressed to use at most $\min\{a,n+2s\}$ ancillas, without increasing the absolute error or $T$-count. For general second-quantized Hamiltonians with bounded one- and two-body coefficients, at operator-norm block-encoding error $ε$, a volume-covering argument combined with circuit counting gives the worst-case lower bound $Ω(n^2\sqrt{\log(n^4/ε)})$, matching the existing upper bound at fixed precision. For the bond-dependent Kitaev honeycomb family on $n$ spins, we obtain independent lower bounds $Ω(n)$ from stabilizer nullity and $Ω(\log(1/ε))$ from one-qubit state preparation, established using different Hamiltonian instances. Together with an explicit LCU construction, they give the tight worst-case scaling $Θ(n+\log(1/ε))$. As an application, we evaluate the $T$-count of a Hamiltonian simulation circuit based on quantum singular value transformation, with each block-encoding query compiled separately. When phase synthesis and controlled queries add at most constant-factor overhead, the simulation $T$-count scales as the query count times the optimal $T$-count per query.

Comments
39 pages, 2 figures

Related papers

$T$-efficient unitary synthesis via block-sparse Hamiltonian simulation

Diyi Liu, Shuchen Zhu, Wibe Albert de Jong +3 more

We construct Clifford+$T$ circuits implementing arbitrary $n$-qubit unitaries with worst-case $T$-count $\widetilde{O}(N^{8/7})$ at constant operator-norm error, where $N=2^n$, improving the previous exponent $5/4$. The construction uses $\widetilde{O}(N^{6/7})$ clean ancillas. The main idea is to reduce the data addressed by each query to the block-encoding circuit for a sparse piece while controlling the additional simulation work needed to combine the pieces. We simulate a Hermitian logarithm of the target by flattening its traceless part with a two-layer random similarity transformation, randomly partitioning the transformed generator into block-sparse Hermitian pieces, and combining their evolutions using qDRIFT. For a target operator-norm error $ε\in(0,1/2)$, let $L=\log_2(N/ε)$, the qubit count plus the requested bits of accuracy. By successively correcting the approximation error, we obtain a circuit using $\widetilde{O}(N^{8/7}L^{3/7}+NL)$ $T$ gates and $\widetilde{O}(N)$ clean ancillas whenever $L\le N$. We also prove that, at constant precision, the exponent $8/7$ is optimal within a sequential block-SELECT model.

quant-pharXiv:2610.18589•Oct 14, 2026•0 citations

Quantum space-depth tradeoffs for coherent block encodings

Yuxin Zhang, Changpeng Shao

Block encodings are a basic interface between quantum algorithms and linear algebra. Standard LCU constructions achieve optimal circuit depth but typically require logarithmically many ancilla qubits. We ask how much quantum workspace can be reduced without sacrificing circuit depth, and study this tradeoff from both algorithmic and lower-bound perspectives. For a Hermitian decomposition $A=\sum_{j=1}^L α_j H_j$, with $\|H_j\|=1$ and $α=\sum_j|α_j|$, we give two coherent $\varepsilon$-approximate block-encoding constructions. The first uses one ancilla qubit and has depth $\widetilde O(L(α/\varepsilon)^{o(1)})$, while the second uses $O(\log\log(α/\varepsilon))$ ancillas and achieves depth $\widetilde O(L)$. For a broad Suzuki-based coherent simulation architecture, we prove an ancilla-depth tradeoff. In the polynomial-resource regime and for a constant number of coherent rounds, $\log(1/\varepsilon)\le O((\log Q)^2+2^a\log Q)$, where $Q$ is depth normalized by the number of Hamiltonian terms and $a$ is the ancilla count. Thus polylogarithmic dependence on $1/\varepsilon$ requires more than constantly many ancillas within this architecture. In a separate repeated-query LCU model, for balanced coefficients $1/L$ and error $\varepsilon=η/L$ with fixed $0<η<1$, we prove $2^a=Ω_η(L^2/(T+L))$, where $T$ is the number of oracle queries. Hence $a=Ω(\log L)$ when $T=O(L^α)$ for some $α<2$. Moreover, in the exact case, $a\ge \log L$ regardless of $T$. We also extend this tradeoff to arbitrary nonnegative coefficients. Finally, we apply our low-ancilla constructions to normalized trace estimation in DQC1, obtaining an optimal algorithm linear in the approximate degree together with a matching query lower bound. Together, these results establish quantitative space-depth and space-query tradeoffs in two natural circuit models.

quant-pharXiv:2607.01843•Jul 3, 2026•0 citations

Nearly Optimal T-Count for Symmetry-Based Quantum State Purification

Jialiang Tang, Junhong Nie, Keming He +2 more

Protecting quantum information from noise is essential for reliable quantum computation and quantum memories. Quantum state purification addresses this challenge by processing multiple noisy copies of an unknown quantum state to produce an output state of higher fidelity. Here we establish nearly tight upper and lower bounds on the non-Clifford resources required to implement the symmetric-projection purification protocol. For $n$ copies of a $d$-dimensional system, we construct a Clifford+T circuit that block-encodes the symmetric projector with operator-norm error at most $ε$ using $\widetilde{O}(n)$ T gates, where the tilde-$O$ notation suppresses polylogarithmic factors in $n$, $d$, and $1/ε$. Conversely, using stabilizer nullity, we prove that any Clifford+T implementation of such a block encoding with error $ε<\frac{1}{2}\sqrt{1-1/n}$ requires at least $2n-2$ T gates, even allowing arbitrary positive normalization and clean stabilizer ancillas. Taken together, these results establish the nearly optimal scaling of non-Clifford cost with the number of copies and uncover the fundamental resource requirements for coherent symmetry projection in quantum state purification.

quant-pharXiv:2610.12098•Oct 9, 2026•0 citations

Low-gate-count block encodings for second-quantized fermionic Hamiltonians

Diyi Liu, Shuchen Zhu, Lin Lin +2 more

Efficient block encoding of many-body Hamiltonians is a central requirement for quantum algorithms in scientific computing, particularly in the early fault-tolerant era. In this work, we introduce new explicit constructions for block encoding second-quantized Hamiltonians that substantially reduce Clifford+T gate complexity and ancilla overhead. By utilizing a data lookup strategy based on the SWAP architecture for the sparsity oracle $O_C$, and a direct sampling method for the amplitude oracle $O_A$ with SELECT-SWAP architecture, we achieve a T count that scales as $\mathcal{\tilde{O}}(\sqrt{L})$ with respect to the number of interaction terms $L$ in general second-quantized Hamiltonians. We also achieve an improved constant factor in the Clifford gate count of our oracle. Furthermore, we design a block encoding that directly targets the $η$-particle subspace, thereby reducing the subnormalization factor from $\mathcal{O}(L)$ to $\mathcal{O}(\sqrt{L})$, and improving fault-tolerant efficiency when simulating systems with fixed particle numbers. Building on the block encoding framework developed for general many-body Hamiltonians, we extend our approach to electronic Hamiltonians whose coefficient tensors exhibit translation invariance or possess decaying structures. Our results provide a practical path toward early fault-tolerant quantum simulation of many-body systems, substantially lowering resource overheads compared to previous methods.

arXiv:2510.08644•Oct 9, 2025•1 citation

Multi-qubit controlled gate synthesis without T-count overhead in the small-error limit

Soichiro Yamazaki, Seiseki Akibue, Kaoru Sano

We study the Clifford+T synthesis of quantum multiplexers $U=\bigoplus_{i=1}^{2^n}U_i$: an $n$-qubit control register selects which single-qubit gate $U_i\in\mathrm{SU}(2)$ acts on the target qubit. For each block $U_i$, consider the minimum even T-count needed to approximate it within diamond distance $\varepsilon$ without ancillae, and let $m$ be the largest of these costs. For an arbitrary multiplexer $U$, we construct an approximation of $U$ within the same distance using at most $m+O(\sqrt{2^n(m+1)})$ $T$ gates. With one control qubit, the cost is $m+O(1)$ and no ancillae are required. We also prove a lower bound that allows the Clifford+T circuit to use any number of clean ancillae initialized to zero and returned exactly to zero. For fixed $n$ and independent Haar-random blocks, the minimum T-count among these circuits lies between $(3-δ)\log_2(1/\varepsilon)$ and $(3+δ)\log_2(1/\varepsilon)$ for every fixed $δ>0$, except with probability $O(\varepsilon^{c_{n,δ}})$ as $\varepsilon\rightarrow0$. The optimal leading coefficient is therefore the same as for a single Haar-random $\mathrm{SU}(2)$ gate. In the lower-bound proof, unitarity forces pairs of off-diagonal blocks describing transitions in opposite directions to satisfy cancellation relations with residuals of order $\varepsilon^2$. We combine these relations with the arithmetic of Clifford+T circuits to bound the Haar measure of targets compatible with each pair of off-diagonal blocks. As an application, we obtain ancilla-free approximations of Haar-random $\mathrm{SU}(4)$ gates within diamond distance $\varepsilon$, using at most $(9+δ)\log_2(1/\varepsilon)$ $T$ gates for every fixed $δ>0$, except with probability $O(\varepsilon^{c_δ})$ with $c_δ>0$.

quant-pharXiv:2603.14202•Mar 17, 2026•0 citations

Quantum Circuit for General Unitary: Improved T-count via Block Flattening and Dilation

Pei Yuan, Shengyu Zhang, Wei Zi

Synthesizing arbitrary $n$-qubit unitaries using as few non-Clifford gates as possible is a central problem in fault-tolerant quantum compilation. We present a Clifford+$T$ quantum circuit construction that approximately implements any classically specified unitary to within error $ε$ and achieves a worst-case $T$-count with leading exponential scaling of $2^{5n/4}$ whenever $\log(1/ε)=\operatorname{poly}(n)$. This improves upon the best previous $2^{4n/3}$ scaling. The key innovation lies in treating the target unitary as a single block-encoded object rather than a long product of simpler operations. A technique of block flattening controls the normalization while preserving an efficient implementation of the block encoding; subsequently, quantum singular value transformation maps its common singular value to one, thereby recovering the target unitary.

quant-pharXiv:2608.17846•Aug 19, 2026•0 citations
We determine the non-Clifford TTT-gate cost of constructing block encodings of structured fermionic and spin Hamiltonians in a unitary Clifford+T+T+T model, when arbitrarily many clean ancillas and unrestricted block-encoding subnormalization are allowed, but without mid-circuit measurements or classical feed-forward. Our main technical tool is an ancilla-compression theorem: any block encoding of an nnn-qubit operator with aaa clean ancillas and at most sss TTT gates can be compressed to use at most min⁡{a,n+2s}\min\{a,n+2s\}min{a,n+2s} ancillas, without increasing the absolute error or TTT-count. For general second-quantized Hamiltonians with bounded one- and two-body coefficients, at operator-norm block-encoding error εεε, a volume-covering argument combined with circuit counting gives the worst-case lower bound Ω(n2log⁡(n4/ε))Ω(n^2\sqrt{\log(n^4/ε)})Ω(n2log(n4/ε)​), matching the existing upper bound at fixed precision. For the bond-dependent Kitaev honeycomb family on nnn spins, we obtain independent lower bounds Ω(n)Ω(n)Ω(n) from stabilizer nullity and Ω(log⁡(1/ε))Ω(\log(1/ε))Ω(log(1/ε)) from one-qubit state preparation, established using different Hamiltonian instances. Together with an explicit LCU construction, they give the tight worst-case scaling Θ(n+log⁡(1/ε))Θ(n+\log(1/ε))Θ(n+log(1/ε)). As an application, we evaluate the TTT-count of a Hamiltonian simulation circuit based on quantum singular value transformation, with each block-encoding query compiled separately. When phase synthesis and controlled queries add at most constant-factor overhead, the simulation TTT-count scales as the query count times the optimal TTT-count per query.
TTT-efficient unitary synthesis via block-sparse Hamiltonian simulation
We construct Clifford+TTT circuits implementing arbitrary nnn-qubit unitaries with worst-case TTT-count O~(N8/7)\widetilde{O}(N^{8/7})O(N8/7) at constant operator-norm error, where N=2nN=2^nN=2n, improving the previous exponent 5/45/45/4. The construction uses O~(N6/7)\widetilde{O}(N^{6/7})O(N6/7) clean ancillas. The main idea is to reduce the data addressed by each query to the block-encoding circuit for a sparse piece while controlling the additional simulation work needed to combine the pieces. We simulate a Hermitian logarithm of the target by flattening its traceless part with a two-layer random similarity transformation, randomly partitioning the transformed generator into block-sparse Hermitian pieces, and combining their evolutions using qDRIFT. For a target operator-norm error ε∈(0,1/2)ε\in(0,1/2)ε∈(0,1/2), let L=log⁡2(N/ε)L=\log_2(N/ε)L=log2​(N/ε), the qubit count plus the requested bits of accuracy. By successively correcting the approximation error, we obtain a circuit using O~(N8/7L3/7+NL)\widetilde{O}(N^{8/7}L^{3/7}+NL)O(N8/7L3/7+NL) TTT gates and O~(N)\widetilde{O}(N)O(N) clean ancillas whenever L≤NL\le NL≤N. We also prove that, at constant precision, the exponent 8/78/78/7 is optimal within a sequential block-SELECT model.
Block encodings are a basic interface between quantum algorithms and linear algebra. Standard LCU constructions achieve optimal circuit depth but typically require logarithmically many ancilla qubits. We ask how much quantum workspace can be reduced without sacrificing circuit depth, and study this tradeoff from both algorithmic and lower-bound perspectives. For a Hermitian decomposition A=∑j=1LαjHjA=\sum_{j=1}^L α_j H_jA=∑j=1L​αj​Hj​, with ∥Hj∥=1\|H_j\|=1∥Hj​∥=1 and α=∑j∣αj∣α=\sum_j|α_j|α=∑j​∣αj​∣, we give two coherent ε\varepsilonε-approximate block-encoding constructions. The first uses one ancilla qubit and has depth O~(L(α/ε)o(1))\widetilde O(L(α/\varepsilon)^{o(1)})O(L(α/ε)o(1)), while the second uses O(log⁡log⁡(α/ε))O(\log\log(α/\varepsilon))O(loglog(α/ε)) ancillas and achieves depth O~(L)\widetilde O(L)O(L). For a broad Suzuki-based coherent simulation architecture, we prove an ancilla-depth tradeoff. In the polynomial-resource regime and for a constant number of coherent rounds, log⁡(1/ε)≤O((log⁡Q)2+2alog⁡Q)\log(1/\varepsilon)\le O((\log Q)^2+2^a\log Q)log(1/ε)≤O((logQ)2+2alogQ), where QQQ is depth normalized by the number of Hamiltonian terms and aaa is the ancilla count. Thus polylogarithmic dependence on 1/ε1/\varepsilon1/ε requires more than constantly many ancillas within this architecture. In a separate repeated-query LCU model, for balanced coefficients 1/L1/L1/L and error ε=η/L\varepsilon=η/Lε=η/L with fixed 0<η<10<η<10<η<1, we prove 2a=Ωη(L2/(T+L))2^a=Ω_η(L^2/(T+L))2a=Ωη​(L2/(T+L)), where TTT is the number of oracle queries. Hence a=Ω(log⁡L)a=Ω(\log L)a=Ω(logL) when T=O(Lα)T=O(L^α)T=O(Lα) for some α<2α<2α<2. Moreover, in the exact case, a≥log⁡La\ge \log La≥logL regardless of TTT. We also extend this tradeoff to arbitrary nonnegative coefficients. Finally, we apply our low-ancilla constructions to normalized trace estimation in DQC1, obtaining an optimal algorithm linear in the approximate degree together with a matching query lower bound. Together, these results establish quantitative space-depth and space-query tradeoffs in two natural circuit models.
Protecting quantum information from noise is essential for reliable quantum computation and quantum memories. Quantum state purification addresses this challenge by processing multiple noisy copies of an unknown quantum state to produce an output state of higher fidelity. Here we establish nearly tight upper and lower bounds on the non-Clifford resources required to implement the symmetric-projection purification protocol. For nnn copies of a ddd-dimensional system, we construct a Clifford+T circuit that block-encodes the symmetric projector with operator-norm error at most εεε using O~(n)\widetilde{O}(n)O(n) T gates, where the tilde-OOO notation suppresses polylogarithmic factors in nnn, ddd, and 1/ε1/ε1/ε. Conversely, using stabilizer nullity, we prove that any Clifford+T implementation of such a block encoding with error ε<121−1/nε<\frac{1}{2}\sqrt{1-1/n}ε<21​1−1/n​ requires at least 2n−22n-22n−2 T gates, even allowing arbitrary positive normalization and clean stabilizer ancillas. Taken together, these results establish the nearly optimal scaling of non-Clifford cost with the number of copies and uncover the fundamental resource requirements for coherent symmetry projection in quantum state purification.
Efficient block encoding of many-body Hamiltonians is a central requirement for quantum algorithms in scientific computing, particularly in the early fault-tolerant era. In this work, we introduce new explicit constructions for block encoding second-quantized Hamiltonians that substantially reduce Clifford+T gate complexity and ancilla overhead. By utilizing a data lookup strategy based on the SWAP architecture for the sparsity oracle OCO_COC​, and a direct sampling method for the amplitude oracle OAO_AOA​ with SELECT-SWAP architecture, we achieve a T count that scales as O~(L)\mathcal{\tilde{O}}(\sqrt{L})O~(L​) with respect to the number of interaction terms LLL in general second-quantized Hamiltonians. We also achieve an improved constant factor in the Clifford gate count of our oracle. Furthermore, we design a block encoding that directly targets the ηηη-particle subspace, thereby reducing the subnormalization factor from O(L)\mathcal{O}(L)O(L) to O(L)\mathcal{O}(\sqrt{L})O(L​), and improving fault-tolerant efficiency when simulating systems with fixed particle numbers. Building on the block encoding framework developed for general many-body Hamiltonians, we extend our approach to electronic Hamiltonians whose coefficient tensors exhibit translation invariance or possess decaying structures. Our results provide a practical path toward early fault-tolerant quantum simulation of many-body systems, substantially lowering resource overheads compared to previous methods.
We study the Clifford+T synthesis of quantum multiplexers U=⨁i=12nUiU=\bigoplus_{i=1}^{2^n}U_iU=⨁i=12n​Ui​: an nnn-qubit control register selects which single-qubit gate Ui∈SU(2)U_i\in\mathrm{SU}(2)Ui​∈SU(2) acts on the target qubit. For each block UiU_iUi​, consider the minimum even T-count needed to approximate it within diamond distance ε\varepsilonε without ancillae, and let mmm be the largest of these costs. For an arbitrary multiplexer UUU, we construct an approximation of UUU within the same distance using at most m+O(2n(m+1))m+O(\sqrt{2^n(m+1)})m+O(2n(m+1)​) TTT gates. With one control qubit, the cost is m+O(1)m+O(1)m+O(1) and no ancillae are required. We also prove a lower bound that allows the Clifford+T circuit to use any number of clean ancillae initialized to zero and returned exactly to zero. For fixed nnn and independent Haar-random blocks, the minimum T-count among these circuits lies between (3−δ)log⁡2(1/ε)(3-δ)\log_2(1/\varepsilon)(3−δ)log2​(1/ε) and (3+δ)log⁡2(1/ε)(3+δ)\log_2(1/\varepsilon)(3+δ)log2​(1/ε) for every fixed δ>0δ>0δ>0, except with probability O(εcn,δ)O(\varepsilon^{c_{n,δ}})O(εcn,δ​) as ε→0\varepsilon\rightarrow0ε→0. The optimal leading coefficient is therefore the same as for a single Haar-random SU(2)\mathrm{SU}(2)SU(2) gate. In the lower-bound proof, unitarity forces pairs of off-diagonal blocks describing transitions in opposite directions to satisfy cancellation relations with residuals of order ε2\varepsilon^2ε2. We combine these relations with the arithmetic of Clifford+T circuits to bound the Haar measure of targets compatible with each pair of off-diagonal blocks. As an application, we obtain ancilla-free approximations of Haar-random SU(4)\mathrm{SU}(4)SU(4) gates within diamond distance ε\varepsilonε, using at most (9+δ)log⁡2(1/ε)(9+δ)\log_2(1/\varepsilon)(9+δ)log2​(1/ε) TTT gates for every fixed δ>0δ>0δ>0, except with probability O(εcδ)O(\varepsilon^{c_δ})O(εcδ​) with cδ>0c_δ>0cδ​>0.
Synthesizing arbitrary nnn-qubit unitaries using as few non-Clifford gates as possible is a central problem in fault-tolerant quantum compilation. We present a Clifford+TTT quantum circuit construction that approximately implements any classically specified unitary to within error εεε and achieves a worst-case TTT-count with leading exponential scaling of 25n/42^{5n/4}25n/4 whenever log⁡(1/ε)=poly⁡(n)\log(1/ε)=\operatorname{poly}(n)log(1/ε)=poly(n). This improves upon the best previous 24n/32^{4n/3}24n/3 scaling. The key innovation lies in treating the target unitary as a single block-encoded object rather than a long product of simpler operations. A technique of block flattening controls the normalization while preserving an efficient implementation of the block encoding; subsequently, quantum singular value transformation maps its common singular value to one, thereby recovering the target unitary.