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arXiv:quant-ph/0211172

Decoherence-Free Subspaces in Supersymmetric Oscillator Networks

Jeffrey Satinover

Published Nov 26, 2002• 0 citations
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Abstract

Quantum superpositions can be used for parallel information processing, but only if protected against decoherence. A two-particle four-state system may have two-dimensional subspaces that are partially or completely decoherence-free, e.g., the symmetric triplet state as an example of the former, the anti-symmetric singlet state of the latter. By extension, a multiparticle system that in the laboratory basis is plagued by decoherence may in some other basis exhibit the symmetries that yield such decoherence-free subspaces (DFS's). Fully-interacting many-fermion spin 1/2 networks may be mathematically transformed to a more tractable many-to-one (or -to-some) variant. This paper applies such a transformation to a hypothetical network of boson-like operators and then argues that a fully-interacting particle number-preserving network of bosons plus fermions with supersymmetric degrees of freedom may be more plausibly exploited so as to contain DFS's. Physical systems that in some basis are inherently anti-symmetric are already known to be useful for quantum information processing. Supersymmetric systems may be likewise.

DOI
10.48550/arxiv.quant-ph/0211172

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Quantum superpositions can be used for parallel information processing, but only if protected against decoherence. A two-particle four-state system may have two-dimensional subspaces that are partially or completely decoherence-free, e.g., the symmetric triplet state as an example of the former, the anti-symmetric singlet state of the latter. By extension, a multiparticle system that in the laboratory basis is plagued by decoherence may in some other basis exhibit the symmetries that yield such decoherence-free subspaces (DFS's). Fully-interacting many-fermion spin 1/2 networks may be mathematically transformed to a more tractable many-to-one (or -to-some) variant. This paper applies such a transformation to a hypothetical network of boson-like operators and then argues that a fully-interacting particle number-preserving network of bosons plus fermions with supersymmetric degrees of freedom may be more plausibly exploited so as to contain DFS's. Physical systems that in some basis are inherently anti-symmetric are already known to be useful for quantum information processing. Supersymmetric systems may be likewise.
A simplified Bogoliubov transform reduces a fully-interacting many-fermion spin-1/2 system-plus-environment to a more tractable many-to-one variant. The transform additionally yields exact solutions for bosonic multi-particle interactions sans the approximation introduced by using discrete time steps to deal with quantum parallelism. The decohering effect of relatively general finite environments is therewith modeled and compared to the decohering effect of an infinite environmental "bath." The anti-symmetric singlet state formed by two maximally-entangled two-state particles is shown to be inherently decoherence-free. As a quantum bit ("qubit") it is thus potentially superior to any single-particle state.
We present a technique to build, within a dissipative bosonic network, decoherence-free channels (DFCs): a group of normal-mode oscillators with null effective damping rates. We verify that the states protected within the DFC define the well-known decoherence-free subspaces (DFSs) when mapped back into the natural network oscillators. Therefore, our technique to build protected normal-mode channels turns out to be an alternative way to build DFSs, which offers advantages over the conventional method. It enables the computation of all the network-protected states at once, as well as leading naturally to the concept of the decoherence quasi-free subspace (DQFS), inside which a superposition state is quasi-completely protected against decoherence. The concept of the DQFS, weaker than that of the DFS, may provide a more manageable mechanism to control decoherence. Finally, as an application of the DQFSs, we show how to build them for quasi-perfect state transfer in networks of coupled quantum dissipative oscillators.
Decoherence-free subspaces and subsystems (DFS) preserve quantum information by encoding it into symmetry-protected states unaffected by decoherence. An inherent DFS of a given experimental system may not exist; however, through the use of dynamical decoupling (DD), one can induce symmetries that support DFSs. Here, we provide the first experimental demonstration of DD-generated decoherence-free subsystem logical qubits. Utilizing IBM Quantum superconducting processors, we investigate two and three-qubit DFS codes comprising up to six and seven noninteracting logical qubits, respectively. Through a combination of DD and error detection, we show that DFS logical qubits can achieve up to a 23% improvement in state preservation fidelity over physical qubits subject to DD alone. This constitutes a beyond-breakeven fidelity improvement for DFS-encoded qubits. Our results showcase the potential utility of DFS codes as a pathway toward enhanced computational accuracy via logical encoding on quantum processors.
Decoherence-free subspace (DFS) provides a crucial mechanism for passive error mitigation in quantum computation by encoding information within symmetry-protected subspaces of the Hilbert space, which are immune from collective decoherence. Constructing a complete set of orthogonal basis states for the DFS is essential to realize fault-tolerant quantum computation by using the DFS codes. However, existing methods for preparing these basis states are often non-scalable, platform-specific, or yield mixed states. Here, we propose a deterministic approach to prepare pure, orthogonal and complete DFS basis states for systems of arbitrary size composed of qubits. Our method employs projective measurements and quantum circuits with single-qubit, two-qubit and Toffoli gates. We provide a rigorous resource cost analysis both mathematically and numerically. Meanwhile, we demonstrate the realizability of our method on NISQ devices by discussing how to implement our method on a superconducting chip. The proposed method offers a universal solution for preparing the DFS basis states across diverse quantum computing platforms and system sizes, which is realizable in the NISQ era.
Evolution of quantum states of array of quantum dots is analyzed by means of numerical solution of the von Neumann equation. For two qubit system with dipole–dipole interaction and common phonon bath the evolution of the symmetric state ↑↓+↓↑2\frac{{ \uparrow \downarrow + \downarrow \uparrow }}{{\sqrt 2 }}2​↑↓+↓↑​ leads to the mixture of the triplet states, leaving the singlet decoupled. For three qubit system (D1/2⊗3=D3/2+2D1/2)\left( {D_{1/2}^{ \otimes 3} = D_{3/2} + 2D_{1/2} } \right)(D1/2⊗3​=D3/2​+2D1/2​) with common phonon bath we observed similar effects within the quartet state D3/2 if all qubits were symmetrically connected.
A general scheme to perform universal, fault-tolerant quantum computation within decoherence-free subspaces (DFSs) is presented. At most two-qubit interactions are required, and the system remains within the DFS throughout the entire implementation of a quantum gate. We show explicitly how to perform universal computation on clusters of the four-qubit DFS encoding one logical qubit each under spatially symmetric (collective) decoherence. Our results have immediate relevance to quantum computer implementations in which quantum logic is implemented through exchange interactions, such as the recently proposed spin-spin coupled quantum dot arrays and donor-atom arrays.