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:0905.1249 · Physical Review Letters

Holonomic Quantum Computation in Subsystems

Ognyan Oreshkov

Published Aug 25, 2009• 24 citations
arXivHTMLPDF

Abstract

We introduce a generalized method of holonomic quantum computation (HQC) based on encoding in subsystems. As an application, we propose a scheme for applying holonomic gates to unencoded qubits by the use of a noisy ancillary qubit. This scheme does not require initialization in a subspace since all dynamical effects factor out as a transformation on the ancilla. We use this approach to show how fault-tolerant HQC can be realized via 2-local Hamiltonians with perturbative gadgets.

DOI
10.1103/physrevlett.103.090502

Related papers

Fault tolerance for holonomic quantum computation

Ognyan Oreshkov, Todd A. Brun, Daniel A. Lidar

In Chapter 16 it was shown how holonomic quantum computation (HQC) can be combined with the method of decoherence-free subspaces (DFSs), leading to passive protection against certain types of correlated errors. However, this is not enough for fault tolerance since other types of errors can accumulate detrimentally unless corrected. Scalability of HQC therefore requires going beyond that scheme, e.g., by combining the holonomic approach with active error correction. One way of combining HQC with active quantum error-correcting codes, which is similar to the way HQC is combined with DFSs, was also mentioned in Chapter 16. This approach, however, is not scalable since it requires Hamiltonians that commute with the stabilizer elements, and when the code increases in size, this necessitates couplings that become increasingly nonlocal. In this chapter, we will show how HQC can be made fault tolerant by combining it with the techniques for fault-tolerant quantum error correction (FTQEC) on stabilizer codes using Hamiltonians of finite locality. The fact that the holonomic method can be mapped directly to the circuit model allows us to construct procedures that resort almost entirely to these techniques. We will discuss an approach that makes use of the encoding already present in a stabilizer code and does not require additional qubits [O08, OBL08]. An alternative approach, which requires ancillary qubits for implementing transversal operations between qubits in the code, can be found in [O09].

Cambridge University Press eBooks•Sep 5, 2013•1 citation

QUANTUM HOLONOMIES FOR QUANTUM COMPUTING

Jiannis K. Pachos, Paolo Zanardi A

Holonomic Quantum Computation (HQC) is an all-geometrical approach to quantum information processing. In the HQC strategy information is encoded in degenerate eigen-spaces of a parametric family of Hamiltonians. The computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops the nontrivial curvature of the underlying bundle geometry gives rise to unitary transformations i.e., holonomies that implement the desired unitary transformations. Conditions necessary for universal QC are stated in terms of the curvature associated to the non-abelian gauge potential (connection) over the control manifold. In view of their geometrical nature the holonomic gates are robust against several kind of perturbations and imperfections. This fact along with the adiabatic fashion in which gates are performed makes in principle HQC an appealing way towards universal fault-tolerant QC.

arXiv:quant-ph/0007110•International Journal of Modern Physics B•Apr 10, 2001•135 citations

Fault-tolerant scheme of holonomic quantum computation on stabilizer codes with robustness to low-weight thermal noise

Yi-Cong Zheng, Todd A. Brun

We show an equivalence relation between fault-tolerant circuits for a stabilizer code and fault-tolerant adiabatic processes for holonomic quantum computation (HQC), in the case where quantum information is encoded in the degenerate ground space of the system Hamiltonian. By this equivalence, we can systematically construct a fault-tolerant HQC scheme, which can geometrically implement a universal set of encoded quantum gates by adiabatically deforming the system Hamiltonian. During this process, quantum information is protected from low-weight thermal excitations by an energy gap that does not change with the problem size.

arXiv:1309.1534•Physical Review A•Mar 12, 2014•13 citations

ddd-pod realization of nonadiabatic holonomic quantum computation

Oskar Axelsson, Claes Fälth, Elias Henriksson Lindberg +1 more

Holonomic quantum computation (HQC) realizes quantum gates through non-Abelian geometric phases, providing an experimentally accessible approach to quantum control. While the nonadiabatic HQC framework has been extensively developed for three-level $Λ$ systems encoding qubits, its systematic extension to higher-dimensional qudits remains largely unexplored. In this work, we generalize nonadiabatic HQC to a $d$-pod configuration, where a single excited state is coupled to $d$ ground states, the latter forming the computational subspace. This scheme enables universal holonomic single- and two-qudit gates using only optical or microwave pulses on trapped atoms or ions, offering an efficient route to implement a discrete universal gate set with minimal pulse coordination. As an explicit example, we analyze in detail the qutrit ($d=3$) case, demonstrating compact realizations of single- and two-qutrit holonomic gates, each gate requiring at most two loops in the Grassmannian generated by at most three pulses.

quant-pharXiv:2609.16216•Sep 16, 2026•0 citations

Expedited Holonomic Quantum Computation in Decoherence-Free Subspace

P. V. Pyshkin, Da-Wei Luo, Jun Jing +2 more

Holonomic quantum computation (HQC) may not show its full potential in quantum speedup due to the prerequisite of a long coherent runtime imposed by the adiabatic condition. Here we show that the conventional HQC can be dramatically accelerated by using external control fields, of which the effectiveness is exclusively determined by the integral of the control fields in the time domain. Remarkably this control scheme can be realized with net zero energy cost and it is fault-tolerant against fluctuation and noise, significantly relaxing the experimental constraints. We demonstrate how to realize the scheme via decoherence-free subspaces. In this way we unify quantum robustness merits of this fault-tolerant control scheme, the conventional HQC and decoherence-free subspace, and propose an {\em expedited} holonomic quantum computation protocol.

arXiv:1507.00815•Jul 3, 2015•2 citations

Scheme for fault-tolerant holonomic computation on stabilizer codes

Ognyan Oreshkov, Todd A. Brun, Daniel A. Lidar

This paper generalizes and expands upon the work [O. Oreshkov, T. A. Brun, and D. A. Lidar, Phys. Rev. Lett. 102, 070502 (2009)] where we introduced a scheme for fault-tolerant holonomic quantum computation (HQC) on stabilizer codes. HQC is an all-geometric strategy based on non-Abelian adiabatic holonomies, which is known to be robust against various types of errors in the control parameters. The scheme we present shows that HQC is a scalable method of computation and opens the possibility for combining the benefits of error correction with the inherent resilience of the holonomic approach. We show that with the Bacon-Shor code the scheme can be implemented using Hamiltonian operators of weights 2 and 3.

arXiv:0904.2143•Physical Review A•Aug 19, 2009•18 citations
We introduce a generalized method of holonomic quantum computation (HQC) based on encoding in subsystems. As an application, we propose a scheme for applying holonomic gates to unencoded qubits by the use of a noisy ancillary qubit. This scheme does not require initialization in a subspace since all dynamical effects factor out as a transformation on the ancilla. We use this approach to show how fault-tolerant HQC can be realized via 2-local Hamiltonians with perturbative gadgets.
In Chapter 16 it was shown how holonomic quantum computation (HQC) can be combined with the method of decoherence-free subspaces (DFSs), leading to passive protection against certain types of correlated errors. However, this is not enough for fault tolerance since other types of errors can accumulate detrimentally unless corrected. Scalability of HQC therefore requires going beyond that scheme, e.g., by combining the holonomic approach with active error correction. One way of combining HQC with active quantum error-correcting codes, which is similar to the way HQC is combined with DFSs, was also mentioned in Chapter 16. This approach, however, is not scalable since it requires Hamiltonians that commute with the stabilizer elements, and when the code increases in size, this necessitates couplings that become increasingly nonlocal. In this chapter, we will show how HQC can be made fault tolerant by combining it with the techniques for fault-tolerant quantum error correction (FTQEC) on stabilizer codes using Hamiltonians of finite locality. The fact that the holonomic method can be mapped directly to the circuit model allows us to construct procedures that resort almost entirely to these techniques. We will discuss an approach that makes use of the encoding already present in a stabilizer code and does not require additional qubits [O08, OBL08]. An alternative approach, which requires ancillary qubits for implementing transversal operations between qubits in the code, can be found in [O09].
Holonomic Quantum Computation (HQC) is an all-geometrical approach to quantum information processing. In the HQC strategy information is encoded in degenerate eigen-spaces of a parametric family of Hamiltonians. The computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops the nontrivial curvature of the underlying bundle geometry gives rise to unitary transformations i.e., holonomies that implement the desired unitary transformations. Conditions necessary for universal QC are stated in terms of the curvature associated to the non-abelian gauge potential (connection) over the control manifold. In view of their geometrical nature the holonomic gates are robust against several kind of perturbations and imperfections. This fact along with the adiabatic fashion in which gates are performed makes in principle HQC an appealing way towards universal fault-tolerant QC.
We show an equivalence relation between fault-tolerant circuits for a stabilizer code and fault-tolerant adiabatic processes for holonomic quantum computation (HQC), in the case where quantum information is encoded in the degenerate ground space of the system Hamiltonian. By this equivalence, we can systematically construct a fault-tolerant HQC scheme, which can geometrically implement a universal set of encoded quantum gates by adiabatically deforming the system Hamiltonian. During this process, quantum information is protected from low-weight thermal excitations by an energy gap that does not change with the problem size.
Holonomic quantum computation (HQC) realizes quantum gates through non-Abelian geometric phases, providing an experimentally accessible approach to quantum control. While the nonadiabatic HQC framework has been extensively developed for three-level ΛΛΛ systems encoding qubits, its systematic extension to higher-dimensional qudits remains largely unexplored. In this work, we generalize nonadiabatic HQC to a ddd-pod configuration, where a single excited state is coupled to ddd ground states, the latter forming the computational subspace. This scheme enables universal holonomic single- and two-qudit gates using only optical or microwave pulses on trapped atoms or ions, offering an efficient route to implement a discrete universal gate set with minimal pulse coordination. As an explicit example, we analyze in detail the qutrit (d=3d=3d=3) case, demonstrating compact realizations of single- and two-qutrit holonomic gates, each gate requiring at most two loops in the Grassmannian generated by at most three pulses.
Holonomic quantum computation (HQC) may not show its full potential in quantum speedup due to the prerequisite of a long coherent runtime imposed by the adiabatic condition. Here we show that the conventional HQC can be dramatically accelerated by using external control fields, of which the effectiveness is exclusively determined by the integral of the control fields in the time domain. Remarkably this control scheme can be realized with net zero energy cost and it is fault-tolerant against fluctuation and noise, significantly relaxing the experimental constraints. We demonstrate how to realize the scheme via decoherence-free subspaces. In this way we unify quantum robustness merits of this fault-tolerant control scheme, the conventional HQC and decoherence-free subspace, and propose an {\em expedited} holonomic quantum computation protocol.
This paper generalizes and expands upon the work [O. Oreshkov, T. A. Brun, and D. A. Lidar, Phys. Rev. Lett. 102, 070502 (2009)] where we introduced a scheme for fault-tolerant holonomic quantum computation (HQC) on stabilizer codes. HQC is an all-geometric strategy based on non-Abelian adiabatic holonomies, which is known to be robust against various types of errors in the control parameters. The scheme we present shows that HQC is a scalable method of computation and opens the possibility for combining the benefits of error correction with the inherent resilience of the holonomic approach. We show that with the Bacon-Shor code the scheme can be implemented using Hamiltonian operators of weights 2 and 3.