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arXiv:quant-ph/0407125

Implementing fanout, parity, and Mod gates via spin exchange interactions

Stephen Fenner, Yong Zhang

Published Jul 16, 2004• 3 citations
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Abstract

We show that, for any n > 0, the Heisenberg interaction among 2n qubits (as spin-1/2 particles) can be used to exactly implement an n-qubit parity gate, which is equivalent in constant depth to an n-qubit fanout gate. Either isotropic or nonisotropic versions of the interaction can be used. We generalize our basic results by showing that any Hamiltonian (acting on suitably encoded logical qubits), whose eigenvalues depend quadratically on the Hamming weight of the logical qubit values, can be used to implement generalized Mod_q gates for any q > 1. This paper is a sequel to quant-ph/0309163, and resolves a question left open in that paper.

DOI
10.48550/arxiv.quant-ph/0407125

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Quantum Fanout and GHZ states using spin-exchange interactions

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We show how the fanout operation on $n$ logical qubits can be implemented via spin-exchange (Heisenberg) interactions between $2n$ physical qubits, together with a physical target qubit and $1$- and $2$-qubit gates in constant depth. We also show that the same interactions can be used to implement Mod_q gates for any $q>1$. These results allow for unequal coupling strengths between physical qubits. This work generalizes an earlier result by Fenner & Zhang [arXiv: quant-ph/0407125], wherein the authors showed similar results assuming all pairwise couplings are equal. The current results give exact conditions on the pairwise couplings that allow for this implementation. Precisely, each logical qubit is encoded into two physical qubits. Couplings between physical qubits encoding the same logical qubit are termed as internal couplings and couplings between the ones encoding different logical qubits are termed as external couplings. We show that for a suitable time $T$ of evolution, the following conditions should hold: a) every external coupling should be an odd integer multiple of $π/2T$; b) every internal coupling should be an integer multiple of $π/T$; and c) the external magnetic strength in $z$-direction should be an integer multiple of $π/T$. Since generalized GHZ (''cat'') states can be created in constant depth using fanout, the same interactions can be used to create these states.

arXiv:2502.10602•Feb 14, 2025•0 citations

Quantum Fanout and GHZ states using spin-exchange interactions*

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We show how the fanout operation on p logical qubits can be implemented via spin-exchange (Heisenberg) interactions between 2p physical qubits, together with a physical target qubit and a constant number of 1- and 2-qubit gates. We also show that the same interactions can be used to implement Modqgates for any q>1. These results allow for unequal coupling strengths between physical qubits. This work strengthens an earlier result by Fenner & Zhang [arXiv:quant-ph/0407125], wherein the authors showed similar results assuming all pairwise couplings are equal. The current results give exact conditions on the pairwise couplings that allow for this implementation. Precisely, each logical qubit is encoded into two physical qubits. Couplings between physical qubits encoding the same logical qubit we call internal couplings and couplings between the ones encoding different logical qubits we call external couplings. We show that for any fixed time T of evolution, the following conditions suffice: a) every external coupling should be an odd integer multiple of π/2T; b) every internal coupling should be an integer multiple of π/T; and c) the external magnetic strength in the z-direction should be an integer multiple of π/T. Since generalized GHZ ("cat") states can be created in constant depth using fanout, the same interactions can be used to create these states.

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Implementing the quantum fanout operation with simple pairwise interactions

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It has been shown that, for even $n$, evolving $n$ qubits according to a Hamiltonian that is the sum of pairwise interactions between the particles, can be used to exactly implement an $(n+1)$-qubit fanout gate using a particular constant-depth circuit~[\href{https://arXiv.org/abs/quant-ph/0309163}{arXiv:quant-ph/0309163}]. However, the coupling coefficients in the Hamiltonian considered in that paper are assumed to be all equal. In this paper, we generalize these results and show that for all $n$, including odd $n$, one can exactly implement an $(n+1)$-qubit parity gate and hence, equivalently in constant depth an $(n+1)$-qubit fanout gate, using a similar Hamiltonian but with unequal couplings, and we give an exact characterization of the constraints that the couplings must satisfy in order for them to be adequate to implement fanout via the same circuit.}{In particular, we show the following:Letting $J_{ij}$ be the coupling strength between the $\ordth{i}$ and $\ordth{j}$ qubits, the set of couplings $\{J_{ij}\}$ is adequate to implement fanout via the circuit above if and only if there exists $J>0$ such that 1. each $J_{ij}$ is an odd integer multiple of $J$, and 2. for each $i$, there are an even number of $j\ne i$ such that $J_{ij}/J \equiv 3\pmod{4}$.

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We show that, for even n, evolving n qubits according to a simple Hamiltonian can be used to exactly implement an (n+1)-qubit parity gate, which is equivalent in constant depth to an (n+1)-qubit fanout gate. We also observe that evolving the Hamiltonian for three qubits results in an inversion-on-three-way-equality gate, which together with single-qubit operations is universal for quantum computation.

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We present a novel implementation of an n-qubit fanout gate using resonance engineering. Our proposed mechanism uses Jaynes-Cummings interactions between multiple qubits and a common harmonic oscillator to realize a fanout gate at the system-level. Our theoretical analysis establishes upper bounds on the gate error, demonstrating linear infidelity scaling in constant time -- a favorable trade-off compared to a conventional CNOT decomposition. To validate the performance of our scheme at large system sizes, we exploit permutation symmetry to reduce the simulation complexity from exponential to polynomial in the number of qubits, enabling simulation up to 100 qubits. The results of this numerical analysis are consistent with our theoretical findings and allow us to characterize the performance well. Our gate will enable faster stabilizer readouts and could provide polynomial speedups in many quantum algorithms.

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Shor's algorithm requires Fanout

Lucas Gretta, Malvika Raj Joshi

Shor's algorithm is a canonical quantum supremacy target whose core operation relies on the Quantum Fourier Transform (QFT). In this note, we resolve an open question of Fang, Fenner, Green, Homer and Zhang from 2006 by showing that approximating QFT in constant depth, for any $n$-qubit modulus, necessarily requires the $n$-qubit Fanout operation. Formally, let $\mathsf{QFT}_q$ be the gate acting on $n = \lceil \log q \rceil$ qubits that computes the QFT under modulus $q$. It is known that any $n$-qubit $\mathsf{QFT}_q$ can be implemented in constant depth using $\mathsf{FANOUT}_n$, i.e. $\mathsf{QFT}_q \in \mathsf{QAC}^0_f$. We prove the converse by using a $\mathsf{QFT}_q$ gate to construct a state of "non-negligible felinity". Consequently, $\mathsf{QFT}_q \in \mathsf{QAC}^0 \iff \mathsf{FANOUT}_n \in \mathsf{QAC^0}$. In the case of $q = 2^n$, such as in Shor's, we approximate $\mathsf{FANOUT}_n$ using a single $\mathsf{QFT}_{2^n}$ gate and $O(1)$ two-qubit local gates, thus tying the feasibility of realizing Shor's algorithm with NISQ circuits to that of Fanout.

quant-pharXiv:2608.06703•Aug 10, 2026•0 citations
We show that, for any n > 0, the Heisenberg interaction among 2n qubits (as spin-1/2 particles) can be used to exactly implement an n-qubit parity gate, which is equivalent in constant depth to an n-qubit fanout gate. Either isotropic or nonisotropic versions of the interaction can be used. We generalize our basic results by showing that any Hamiltonian (acting on suitably encoded logical qubits), whose eigenvalues depend quadratically on the Hamming weight of the logical qubit values, can be used to implement generalized Mod_q gates for any q > 1. This paper is a sequel to quant-ph/0309163, and resolves a question left open in that paper.
We show how the fanout operation on nnn logical qubits can be implemented via spin-exchange (Heisenberg) interactions between 2n2n2n physical qubits, together with a physical target qubit and 111- and 222-qubit gates in constant depth. We also show that the same interactions can be used to implement Mod_q gates for any q>1q>1q>1. These results allow for unequal coupling strengths between physical qubits. This work generalizes an earlier result by Fenner & Zhang [arXiv: quant-ph/0407125], wherein the authors showed similar results assuming all pairwise couplings are equal. The current results give exact conditions on the pairwise couplings that allow for this implementation. Precisely, each logical qubit is encoded into two physical qubits. Couplings between physical qubits encoding the same logical qubit are termed as internal couplings and couplings between the ones encoding different logical qubits are termed as external couplings. We show that for a suitable time TTT of evolution, the following conditions should hold: a) every external coupling should be an odd integer multiple of π/2Tπ/2Tπ/2T; b) every internal coupling should be an integer multiple of π/Tπ/Tπ/T; and c) the external magnetic strength in zzz-direction should be an integer multiple of π/Tπ/Tπ/T. Since generalized GHZ (''cat'') states can be created in constant depth using fanout, the same interactions can be used to create these states.
We show how the fanout operation on p logical qubits can be implemented via spin-exchange (Heisenberg) interactions between 2p physical qubits, together with a physical target qubit and a constant number of 1- and 2-qubit gates. We also show that the same interactions can be used to implement Modqgates for any q>1. These results allow for unequal coupling strengths between physical qubits. This work strengthens an earlier result by Fenner & Zhang [arXiv:quant-ph/0407125], wherein the authors showed similar results assuming all pairwise couplings are equal. The current results give exact conditions on the pairwise couplings that allow for this implementation. Precisely, each logical qubit is encoded into two physical qubits. Couplings between physical qubits encoding the same logical qubit we call internal couplings and couplings between the ones encoding different logical qubits we call external couplings. We show that for any fixed time T of evolution, the following conditions suffice: a) every external coupling should be an odd integer multiple of π/2T; b) every internal coupling should be an integer multiple of π/T; and c) the external magnetic strength in the z-direction should be an integer multiple of π/T. Since generalized GHZ ("cat") states can be created in constant depth using fanout, the same interactions can be used to create these states.
It has been shown that, for even nnn, evolving nnn qubits according to a Hamiltonian that is the sum of pairwise interactions between the particles, can be used to exactly implement an (n+1)(n+1)(n+1)-qubit fanout gate using a particular constant-depth circuit~[\href{https://arXiv.org/abs/quant-ph/0309163}{arXiv:quant-ph/0309163}]. However, the coupling coefficients in the Hamiltonian considered in that paper are assumed to be all equal. In this paper, we generalize these results and show that for all nnn, including odd nnn, one can exactly implement an (n+1)(n+1)(n+1)-qubit parity gate and hence, equivalently in constant depth an (n+1)(n+1)(n+1)-qubit fanout gate, using a similar Hamiltonian but with unequal couplings, and we give an exact characterization of the constraints that the couplings must satisfy in order for them to be adequate to implement fanout via the same circuit.}{In particular, we show the following:Letting JijJ_{ij}Jij​ be the coupling strength between the $\ordth{i}$ and $\ordth{j}$ qubits, the set of couplings {Jij}\{J_{ij}\}{Jij​} is adequate to implement fanout via the circuit above if and only if there exists J>0J>0J>0 such that 1. each JijJ_{ij}Jij​ is an odd integer multiple of JJJ, and 2. for each iii, there are an even number of j≠ij\ne ij=i such that Jij/J≡3(mod4)J_{ij}/J \equiv 3\pmod{4}Jij​/J≡3(mod4).
We present a novel implementation of an n-qubit fanout gate using resonance engineering. Our proposed mechanism uses Jaynes-Cummings interactions between multiple qubits and a common harmonic oscillator to realize a fanout gate at the system-level. Our theoretical analysis establishes upper bounds on the gate error, demonstrating linear infidelity scaling in constant time -- a favorable trade-off compared to a conventional CNOT decomposition. To validate the performance of our scheme at large system sizes, we exploit permutation symmetry to reduce the simulation complexity from exponential to polynomial in the number of qubits, enabling simulation up to 100 qubits. The results of this numerical analysis are consistent with our theoretical findings and allow us to characterize the performance well. Our gate will enable faster stabilizer readouts and could provide polynomial speedups in many quantum algorithms.
Shor's algorithm is a canonical quantum supremacy target whose core operation relies on the Quantum Fourier Transform (QFT). In this note, we resolve an open question of Fang, Fenner, Green, Homer and Zhang from 2006 by showing that approximating QFT in constant depth, for any nnn-qubit modulus, necessarily requires the nnn-qubit Fanout operation. Formally, let QFTq\mathsf{QFT}_qQFTq​ be the gate acting on n=⌈log⁡q⌉n = \lceil \log q \rceiln=⌈logq⌉ qubits that computes the QFT under modulus qqq. It is known that any nnn-qubit QFTq\mathsf{QFT}_qQFTq​ can be implemented in constant depth using FANOUTn\mathsf{FANOUT}_nFANOUTn​, i.e. QFTq∈QACf0\mathsf{QFT}_q \in \mathsf{QAC}^0_fQFTq​∈QACf0​. We prove the converse by using a QFTq\mathsf{QFT}_qQFTq​ gate to construct a state of "non-negligible felinity". Consequently, QFTq∈QAC0  ⟺  FANOUTn∈QAC0\mathsf{QFT}_q \in \mathsf{QAC}^0 \iff \mathsf{FANOUT}_n \in \mathsf{QAC^0}QFTq​∈QAC0⟺FANOUTn​∈QAC0. In the case of q=2nq = 2^nq=2n, such as in Shor's, we approximate FANOUTn\mathsf{FANOUT}_nFANOUTn​ using a single QFT2n\mathsf{QFT}_{2^n}QFT2n​ gate and O(1)O(1)O(1) two-qubit local gates, thus tying the feasibility of realizing Shor's algorithm with NISQ circuits to that of Fanout.