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:quant-ph/0506270 · Physical Review A

Spin- 1∕2 particles moving on a two-dimensional lattice with nearest-neighbor interactions can realize an autonomous quantum computer

Dominik Janzing

Published Jan 9, 2007• 27 citations
arXivHTMLPDF

Abstract

What is the simplest Hamiltonian which can implement quantum computation without requiring any control operations during the computation process? In a previous paper we have constructed a ten-local finite-range interaction among qubits on a two-dimensional (2D) lattice having this property. Here we show that pair-interactions among qutrits on a 2D lattice are sufficient, too, and can also implement an ergodic computer where the result can be read out from the time average state after some postselection with high success probability. Two of the three qutrit states are given by the two levels of a spin-$1∕2$ particle located at a specific lattice site, the third state is its absence. Usual hopping terms together with an attractive force among adjacent particles induce a coupled quantum walk where the particle spins are subjected to spatially inhomogeneous interactions implementing holonomic quantum computing. The holonomic method ensures that the implemented circuit does not depend on the time needed for the walk. Even though the implementation of the required type of spin-spin interactions is currently unclear, the model shows that quite simple Hamiltonians are powerful enough to allow for universal quantum computing in a closed physical system.

DOI
10.1103/physreva.75.012307

Related papers

Adiabatic and Hamiltonian computing on a 2D lattice with simple two-qubit interactions

Barbara M. Terhal, Seth Lloyd

We show how to perform universal Hamiltonian and adiabatic computing using a time-independent Hamiltonian on a 2D grid describing a system of hopping particles which string together and interact to perform the computation. In this construction, the movement of one particle is controlled by the presence or absence of other particles, an effective quantum field effect transistor that allows the construction of controlled-NOT and controlled-rotation gates. The construction translates into a model for universal quantum computation with time-independent two-qubit ZZ and XX+YY interactions on an (almost) planar grid. The effective Hamiltonian is arrived at by a single use of first-order perturbation theory avoiding the use of perturbation gadgets. The dynamics and spectral properties of the effective Hamiltonian can be fully determined as it corresponds to a particular realization of a mapping between a quantum circuit and a Hamiltonian called the space–time circuit-to-Hamiltonian construction. Because of the simple interactions required, and because no higher-order perturbation gadgets are employed, our construction is potentially realizable using superconducting or other solid-state qubits.

Institute of Physics eBooks•Feb 1, 2016•1 citation

Adiabatic and Hamiltonian computing on a 2D lattice with simple two-qubit interactions

Seth Lloyd, Barbara M. Terhal

We show how to perform universal Hamiltonian and adiabatic computing using a time-independent Hamiltonian on a 2D grid describing a system of hopping particles which string together and interact to perform the computation. In this construction, the movement of one particle is controlled by the presence or absence of other particles, an effective quantum field effect transistor that allows the construction of controlled-NOT and controlled-rotation gates. The construction translates into a model for universal quantum computation with time-independent 2-qubit ZZ and XX+YY interactions on an (almost) planar grid. The effective Hamiltonian is arrived at by a single use of first-order perturbation theory avoiding the use of perturbation gadgets. The dynamics and spectral properties of the effective Hamiltonian can be fully determined as it corresponds to a particular realization of a mapping between a quantum circuit and a Hamiltonian called the space-time circuit-to-Hamiltonian construction. Because of the simple interactions required, and because no higher-order perturbation gadgets are employed, our construction is potentially realizable using superconducting or other solid-state qubits.

•Jan 1, 2016•22 citations

Adiabatic and Hamiltonian computing on a 2D lattice with simple two-qubit interactions

Seth Lloyd, Barbara M. Terhal

We show how to perform universal Hamiltonian and adiabatic computing using a time-independent Hamiltonian on a 2D grid describing a system of hopping particles which string together and interact to perform the computation. In this construction, the movement of one particle is controlled by the presence or absence of other particles, an effective quantum field effect transistor that allows the construction of controlled-NOT and controlled-rotation gates. The construction translates into a model for universal quantum computation with time-independent two-qubit ZZ and XX + YY interactions on an (almost) planar grid. The effective Hamiltonian is arrived at by a single use of first-order perturbation theory avoiding the use of perturbation gadgets. The dynamics and spectral properties of the effective Hamiltonian can be fully determined as it corresponds to a particular realization of a mapping between a quantum circuit and a Hamiltonian called the space–time circuit-to-Hamiltonian construction. Because of the simple interactions required, and because no higher-order perturbation gadgets are employed, our construction is potentially realizable using superconducting or other solid-state qubits.

arXiv:1509.01278•New Journal of Physics•Feb 12, 2016•23 citations

Hamiltonian quantum cellular automata in one dimension

Daniel Nagaj, Paweł Wocjan

We construct a simple translationally invariant, nearest-neighbor Hamiltonian on a chain of ten-dimensional qudits that makes it possible to realize universal quantum computing without any external control during the computational process. We only require the ability to prepare an initial computational basis state that encodes both the quantum circuit and its input. The computational process is then carried out by the autonomous Hamiltonian time evolution. After a time polynomially long in the size of the quantum circuit has passed, the result of the computation is obtained with high probability by measuring a few qudits in the computational basis. This result also implies that there cannot exist efficient classical simulation methods for generic translationally invariant nearest-neighbor Hamiltonians on qudit chains, unless quantum computers can be efficiently simulated by classical computers (or, put in complexity theoretic terms, unless $\mathrm{BPP}=\mathrm{BQP}$).

arXiv:0802.0886•Physical Review A•Sep 9, 2008•55 citations

Universal quantum computing in linear nearest neighbor architectures

Preethika Kumar, Steven R. Skinner

We introduce a scheme for realizing universal quantum computing in a linear nearest neighbor architecture with fixed couplings. We first show how to realize a controlled-NOT gate operation between two adjacent qubits without having to isolate the two qubits from qubits adjacent to them. The gate operation is implemented by applying two consecutive pulses of equal duration, but varying amplitudes, on the target qubit. Since only a single control parameter is required in implementing our scheme, it is very efficient. We next show how our scheme can be used to realize single qubit rotations and two-qubit controlled-unitary operations. As most proposals for solid state implementations of a quantum computer use a one-dimensional line of qubits, the schemes presented here will be extremely useful.

Quantum Information and Computation•Mar 1, 2011•6 citations

Universal quantum walks and adiabatic algorithms by 1D Hamiltonians

Bradley A. Chase, Andrew J. Landahl

We construct a family of time-independent nearest-neighbor Hamiltonians coupling eight-state systems on a 1D ring that enables universal quantum computation. Hamiltonians in this family can achieve universality either by driving a continuous-time quantum walk or by terminating an adiabatic algorithm. In either case, the universality property can be understood as arising from an efficient simulation of a programmable quantum circuit. Using gadget perturbation theory, one can demonstrate the same kind of universality for related Hamiltonian families acting on qubits in 2D. Our results demonstrate that simulating 1D chains of spin-7/2 particles is BQP-hard, and indeed BQP-complete because the outputs of decision problems can be encoded in the outputs of such simulations.

arXiv:0802.1207•Feb 8, 2008•16 citations
What is the simplest Hamiltonian which can implement quantum computation without requiring any control operations during the computation process? In a previous paper we have constructed a ten-local finite-range interaction among qubits on a two-dimensional (2D) lattice having this property. Here we show that pair-interactions among qutrits on a 2D lattice are sufficient, too, and can also implement an ergodic computer where the result can be read out from the time average state after some postselection with high success probability. Two of the three qutrit states are given by the two levels of a spin-1∕21∕21∕2 particle located at a specific lattice site, the third state is its absence. Usual hopping terms together with an attractive force among adjacent particles induce a coupled quantum walk where the particle spins are subjected to spatially inhomogeneous interactions implementing holonomic quantum computing. The holonomic method ensures that the implemented circuit does not depend on the time needed for the walk. Even though the implementation of the required type of spin-spin interactions is currently unclear, the model shows that quite simple Hamiltonians are powerful enough to allow for universal quantum computing in a closed physical system.
We show how to perform universal Hamiltonian and adiabatic computing using a time-independent Hamiltonian on a 2D grid describing a system of hopping particles which string together and interact to perform the computation. In this construction, the movement of one particle is controlled by the presence or absence of other particles, an effective quantum field effect transistor that allows the construction of controlled-NOT and controlled-rotation gates. The construction translates into a model for universal quantum computation with time-independent two-qubit ZZ and XX+YY interactions on an (almost) planar grid. The effective Hamiltonian is arrived at by a single use of first-order perturbation theory avoiding the use of perturbation gadgets. The dynamics and spectral properties of the effective Hamiltonian can be fully determined as it corresponds to a particular realization of a mapping between a quantum circuit and a Hamiltonian called the space–time circuit-to-Hamiltonian construction. Because of the simple interactions required, and because no higher-order perturbation gadgets are employed, our construction is potentially realizable using superconducting or other solid-state qubits.
We show how to perform universal Hamiltonian and adiabatic computing using a time-independent Hamiltonian on a 2D grid describing a system of hopping particles which string together and interact to perform the computation. In this construction, the movement of one particle is controlled by the presence or absence of other particles, an effective quantum field effect transistor that allows the construction of controlled-NOT and controlled-rotation gates. The construction translates into a model for universal quantum computation with time-independent 2-qubit ZZ and XX+YY interactions on an (almost) planar grid. The effective Hamiltonian is arrived at by a single use of first-order perturbation theory avoiding the use of perturbation gadgets. The dynamics and spectral properties of the effective Hamiltonian can be fully determined as it corresponds to a particular realization of a mapping between a quantum circuit and a Hamiltonian called the space-time circuit-to-Hamiltonian construction. Because of the simple interactions required, and because no higher-order perturbation gadgets are employed, our construction is potentially realizable using superconducting or other solid-state qubits.
We show how to perform universal Hamiltonian and adiabatic computing using a time-independent Hamiltonian on a 2D grid describing a system of hopping particles which string together and interact to perform the computation. In this construction, the movement of one particle is controlled by the presence or absence of other particles, an effective quantum field effect transistor that allows the construction of controlled-NOT and controlled-rotation gates. The construction translates into a model for universal quantum computation with time-independent two-qubit ZZ and XX + YY interactions on an (almost) planar grid. The effective Hamiltonian is arrived at by a single use of first-order perturbation theory avoiding the use of perturbation gadgets. The dynamics and spectral properties of the effective Hamiltonian can be fully determined as it corresponds to a particular realization of a mapping between a quantum circuit and a Hamiltonian called the space–time circuit-to-Hamiltonian construction. Because of the simple interactions required, and because no higher-order perturbation gadgets are employed, our construction is potentially realizable using superconducting or other solid-state qubits.
We construct a simple translationally invariant, nearest-neighbor Hamiltonian on a chain of ten-dimensional qudits that makes it possible to realize universal quantum computing without any external control during the computational process. We only require the ability to prepare an initial computational basis state that encodes both the quantum circuit and its input. The computational process is then carried out by the autonomous Hamiltonian time evolution. After a time polynomially long in the size of the quantum circuit has passed, the result of the computation is obtained with high probability by measuring a few qudits in the computational basis. This result also implies that there cannot exist efficient classical simulation methods for generic translationally invariant nearest-neighbor Hamiltonians on qudit chains, unless quantum computers can be efficiently simulated by classical computers (or, put in complexity theoretic terms, unless BPP=BQP\mathrm{BPP}=\mathrm{BQP}BPP=BQP).
We introduce a scheme for realizing universal quantum computing in a linear nearest neighbor architecture with fixed couplings. We first show how to realize a controlled-NOT gate operation between two adjacent qubits without having to isolate the two qubits from qubits adjacent to them. The gate operation is implemented by applying two consecutive pulses of equal duration, but varying amplitudes, on the target qubit. Since only a single control parameter is required in implementing our scheme, it is very efficient. We next show how our scheme can be used to realize single qubit rotations and two-qubit controlled-unitary operations. As most proposals for solid state implementations of a quantum computer use a one-dimensional line of qubits, the schemes presented here will be extremely useful.
We construct a family of time-independent nearest-neighbor Hamiltonians coupling eight-state systems on a 1D ring that enables universal quantum computation. Hamiltonians in this family can achieve universality either by driving a continuous-time quantum walk or by terminating an adiabatic algorithm. In either case, the universality property can be understood as arising from an efficient simulation of a programmable quantum circuit. Using gadget perturbation theory, one can demonstrate the same kind of universality for related Hamiltonian families acting on qubits in 2D. Our results demonstrate that simulating 1D chains of spin-7/2 particles is BQP-hard, and indeed BQP-complete because the outputs of decision problems can be encoded in the outputs of such simulations.