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arXiv:quant-ph/0407241 · Physical Review Letters

Scalable Fault-Tolerant Quantum Computation in Decoherence-Free Subspaces

Zheng-Wei Zhou, Bo Hao Yu, Xingxiang Zhou, M. J. Feldman, Guang‐Can Guo

Published Jun 1, 2004• 16 citations
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Abstract

We investigate how to concatenate different decoherence-free subspaces to realize scalable universal fault-tolerant quantum computation. Based on tunable $XXZ$ interactions, we present an architecture for scalable quantum computers which can fault-tolerantly perform universal quantum computation by manipulating only a single type of parameter. By using the concept of interaction-free subspaces, we eliminate the need to tune the couplings between logical qubits, which further reduces the technical difficulties for implementing quantum computation.

DOI
10.1103/physrevlett.93.010501

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We investigate how to concatenate different decoherence-free subspaces to realize scalable universal fault-tolerant quantum computation. Based on tunable XXZXXZXXZ interactions, we present an architecture for scalable quantum computers which can fault-tolerantly perform universal quantum computation by manipulating only a single type of parameter. By using the concept of interaction-free subspaces, we eliminate the need to tune the couplings between logical qubits, which further reduces the technical difficulties for implementing quantum computation.
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.
Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this stabilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computation which always preserves the DFS (natural fault-tolerant computation) can be performed using only two-body interactions. This is in marked contrast to standard error correcting codes, where all known constructions using one- or two-body interactions must leave the code space during the on-time of the fault-tolerant gates. A further consequence of our universality construction is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many quantum systems, is thus asymptotically universal.
Decoherence-free subspaces (DFSs) are constructed without the assumption of spatially symmetric system-bath coupling. Instead the underlying assumption is that subgroups of the full Pauli group of errors are responsible for the decoherence. The corresponding decoherence-free states can protect quantum information in the presence of multiple-qubit errors, and are stabilizer codes. It is shown how to perform universal fault tolerant quantum computation on this class of DFSs. This is the first demonstration that it is possible to use only one- and two-body quantum gates to perform full-blown quantum computation on a class of DFSs, with a finite number of measurements.
We present a universal fault-tolerant quantum computing architecture based on identical particle qubits (IPQs), where we find that the first-order IPQ - bath interaction fundamentally differs from the conventional first-order qubit-bath interaction. This key distinction necessitates a redesign of existing strategies to fight decoherence. We propose that the simplest quantum error correction code can be realized directly within the physical qubit, provided that conventional correction and restoration are generalized beyond unitary operations to employ physically implementable reversal operations -- naturally placing logical and physical qubits on equal footing. We further demonstrate that dynamical decoupling (DD) remains effective within this unified framework, and that a decoherence-free subspace (DFS) -- like structure emerges. Unlike previous approximate treatments, our analytically solvable IPQ-Bath model enables rigorous testing of these strategies, with numerical simulations validating their effectiveness.
Most quantum computer realizations require the ability to apply local fields and tune the couplings between qubits, in order to realize single bit and two bit gates which are necessary for universal quantum computation. We present a scheme to remove the necessity of switching the couplings between qubits for two bit gates, which are more costly in many cases. Our strategy is to compute with encoded qubits in and out of carefully designed interaction free subspaces analogous to decoherence free subspaces. We give two examples to show how universal quantum computation is realized in our scheme with local manipulations to physical qubits only, for both diagonal and off diagonal interactions.
Reliable execution of large-scale quantum algorithms requires robust underlying operations and this challenge is addressed by quantum error correction (QEC). Most modern QEC protocols rely on measurements and feed-forward operations, which are experimentally demanding, and often slow and prone to high error rates. Additionally, no single error-correcting code intrinsically supports the full set of logical operations required for universal quantum computing, resulting in an increased operational overhead. In this work, we present a complete toolbox for fault-tolerant universal quantum computing without the need for measurements during algorithm execution by combining the strategies of code switching and concatenation. To this end, we develop new fault-tolerant, measurement-free protocols to transfer encoded information between 2D and 3D color codes, which offer complementary and in combination universal sets of robust logical gates. We identify experimentally realistic regimes where these protocols surpass state-of-the-art measurement-based approaches. Moreover, we extend the scheme to higher-distance codes by concatenating the 2D color code with itself and by integrating code switching for operations that lack a natively fault-tolerant implementation. Our measurement-free approach thereby provides a practical and scalable pathway for universal quantum computing on state-of-the-art quantum processors.