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.
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.
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.
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 Jij be the coupling strength between the $\ordth{i}$ and $\ordth{j}$ qubits, the set of couplings {Jij} is adequate to implement fanout via the circuit above if and only if there exists J>0 such that 1. each Jij is an odd integer multiple of J, and 2. for each i, there are an even number of j=i such that Jij/J≡3(mod4).
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.
Pairwise exchange couplings have long been the standard mechanism for entangling spin qubits in semiconductor systems. However, implementing quantum circuits based on pairwise exchange gates often requires a lengthy sequence of elementary gate operations. In this work, we present an alternative approach: multi-qubit entangling gate operations that simultaneously drive the exchange couplings between multiple pairs of spin qubits. We explore three spin qubit systems in linear or triangular configurations. We derive analytical expressions for these multi-exchange entangling operations and demonstrate how to use the resulting three-qubit gates to construct quantum circuits capable of generating standard entangled states such as GHZ and W states, and the Toffoli gate, by optimizing control parameters. Our results show that this multi-qubit strategy significantly reduces the number of required operations, offering a pathway to more efficient, shallower, and more coherent circuits for spin-qubit processors.
A constructive scheme has been devised to enable mapping of any quantum computation into a spintronic circuit in which the computation is encoded in a basis that is, in principle, immune to quantum decoherence. The scheme is implemented by an algorithm that utilizes multiple physical spins to encode each logical bit in such a way that collective errors affecting all the physical spins do not disturb the logical bit. The scheme is expected to be of use to experimenters working on spintronic implementations of quantum logic. Spintronic computing devices use quantum-mechanical spins (typically, electron spins) to encode logical bits. Bits thus encoded (denoted qubits) are potentially susceptible to errors caused by noise and decoherence. The traditional model of quantum computation is based partly on the assumption that each qubit is implemented by use of a single two-state quantum system, such as an electron or other spin-1.2 particle. It can be surprisingly difficult to achieve certain gate operations . most notably, those of arbitrary 1-qubit gates . in spintronic hardware according to this model. However, ironically, certain 2-qubit interactions (in particular, spin-spin exchange interactions) can be achieved relatively easily in spintronic hardware. Therefore, it would be fortunate if it were possible to implement any 1-qubit gate by use of a spin-spin exchange interaction. While such a direct representation is not possible, it is possible to achieve an arbitrary 1-qubit gate indirectly by means of a sequence of four spin-spin exchange interactions, which could be implemented by use of four exchange gates. Accordingly, the present scheme provides for mapping any 1-qubit gate in the logical basis into an equivalent sequence of at most four spin-spin exchange interactions in the physical (encoded) basis. The complexity of the mathematical derivation of the scheme from basic quantum principles precludes a description within this article; it must suffice to report that the derivation provides explicit constructions for finding the exchange couplings in the physical basis needed to implement any arbitrary 1-qubit gate. These constructions lead to spintronic encodings of quantum logic that are more efficient than those of a previously published scheme that utilizes a universal but fixed set of gates.
We study a resource-efficient approach for implementing logical fanout operations in fault-tolerant distributed quantum computing using transversal operations on quantum error-correcting code blocks. Logical fanout, comprising multiple controlled-NOT operations from a common control qubit to target qubits located at remote nodes, is an important primitive for distributed quantum computation but can require substantial non-local communication when implemented directly between encoded blocks. We exploit the structure of encoded blocks and the availability of transversal logical operations to construct distributed fanout circuits that reduce the required non-local operations while preserving the logical action of the fanout operation. The construction is developed for encoded quantum information and illustrated using Bivariate Bicycle (BB)-code blocks. We analyze the resulting physical gate, entanglement, circuit-depth, and ancilla requirements. The approach provides a systematic method for implementing large logical fanout operations across distributed error-corrected quantum processors. Also, we study a distributed implementation of the global gate GCZ involving logical qubits (encoded using BB-code blocks), exploiting the concurrency in transversal distributed fanouts.