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arXiv:quant-ph/0408162 · Journal of the Optical Society of America B

Algorithm-based analysis of collective decoherence in quantum computation

Shoko Utsunomiya, Cyrus P. Master, Y. Yamamoto

Published Jan 26, 2007• 6 citations
arXivHTMLPDF

Abstract

In a quantum computer, qubits are often stored in identical two-level systems separated by a distance shorter than the characteristic wavelength of the reservoirs that are responsible for decoherence. In this case the collective qubit-reservoir interaction, rather than the individual qubit-reservoir interaction, may determine the decoherence properties. We study the collective decoherence behavior in between each step in certain quantum algorithms and propose a simple alternative of implementing quantum algorithms using a quantum trajectory that is close to a decoherence-free subspace that avoids unstable Dicke's superradiant states and Schrödinger's cat state.

DOI
10.1364/josab.24.000198

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In a quantum computer, qubits are often stored in identical two-level systems separated by a distance shorter than the characteristic wavelength of the reservoirs that are responsible for decoherence. In this case the collective qubit-reservoir interaction, rather than the individual qubit-reservoir interaction, may determine the decoherence properties. We study the collective decoherence behavior in between each step in certain quantum algorithms and propose a simple alternative of implementing quantum algorithms using a quantum trajectory that is close to a decoherence-free subspace that avoids unstable Dicke's superradiant states and Schrödinger's cat state.
The dynamical evolution of a quantum register of arbitrary length coupled to an environment of arbitrary coherence length is predicted within a relevant model of decoherence. The results are reported for quantum bits (qubits) coupling individually to different environments (``independent decoherence'') and qubits interacting collectively with the same reservoir (``collective decoherence''). In both cases, explicit decoherence functions are derived for any number of qubits. The decay of the coherences of the register is shown to strongly depend on the input states: We show that this sensitivity is a characteristic of both types of coupling (collective and independent) and not only of the collective coupling, as has been reported previously. A nontrivial behavior (``recoherence'') is found in the decay of the off-diagonal elements of the reduced density matrix in the specific situation of independent decoherence. Our results lead to the identification of decoherence-free states in the collective decoherence limit. These states belong to subspaces of the system's Hilbert space that do not get entangled with the environment, making them ideal elements for the engineering of ``noiseless'' quantum codes. We also discuss the relations between decoherence of the quantum register and computational complexity based on the dynamical results obtained for the register density matrix.
The idea of coding the information in decoherence-free subspaces (DFS) to avoid decoherence in quantum computations has been receiving a lot of attention in circuit-based quantum computations, since first proposed by Zanardi & Rasetti [1] (Phys. Rev. Lett., 1997). Following [2] (Altaisky & Kaputkina, JISST, 2025), we study the possibility to harness DFS to quantum machine learning, where no explicit algorithm is present, but the quantum system learns to classify data. We have shown that a simple classification problem of distinguishing between 2-qubit product states and 2-qubit entangled states can be solved by 4-qubit quantum reservoir [3] (Ghosh, et al., npj Quantum Inf., 2019), the output of which we fed to decoherence-free subspaces.
A major challenge in the implementation of quantum computers is mitigating the impact of undesired couplings with the environment that lead to decoherence.Here, we discuss an error-avoiding strategy that is based upon freedoms inherent in decomposing a quantum algorithm into fundamental gates.Although this approach is specific to a particular algorithm and a collective noise model, it does not the require complicated encodings or significant additional qubit resources of more general error-correcting codes.This paper also includes an overview of quantum computation, including a discussion of the quantum search algorithm.
Quantum computations have so far proved to be more powerful than classical computations, but quantum computers still have not been put into practical use due to several technical issues. One of the most serious problems for realizing quantum computers is decoherence that occurs inevitably since our apparatus are surrounded with environment and open systems. In this paper, we give some surveys on a variety of effects of decoherence in quantum algorithms such as Grover's database search and quantum walks, and we show how quantum algorithms work under decoherence, how sensitive they are against decoherence, and how to implement a robust quantum circuit.
The Dicke model is a paradigmatic quantum-optical model describing the interaction of a collection of two-level systems with a single bosonic mode. Effective implementations of this model made it possible to observe the emergence of superradiance, i.e., cooperative phenomena arising from the collective nature of light-matter interactions. Via reservoir engineering and analogue quantum simulation techniques, current experimental platforms allow us not only to implement the Dicke model but also to design more exotic interactions, such as the two-photon Dicke model. In the Hamiltonian case, this model presents an interesting phase diagram characterized by two quantum criticalities: a superradiant phase transition and a spectral collapse, that is, the coalescence of discrete energy levels into a continuous band. Here, we investigate the effects of both qubit and photon dissipation on the phase transition and on the instability induced by the spectral collapse. Using a mean-field decoupling approximation, we analytically obtain the steady-state expectation values of the observables signaling a symmetry breaking, identifying a first-order phase transition from the normal to the superradiant phase. Our stability analysis unveils a very rich phase diagram, which features stable, bistable, and unstable phases depending on the dissipation rate.
When an environmental disturbance to a quantum system has a wavelength much larger than the system size, all qubits in the system are under the action of the same error operator. The noiseless subsystem and decoherence-free subspace are immune to such collective noise. We construct simple quantum circuits that implement these error-avoiding codes for a small number nnn of physical qubits. A single logical qubit is encoded with n=3n=3n=3 and 4, while two and three logical qubits are encoded with n=5n=5n=5 and 7, respectively. Recursive relations among subspaces employed in these codes play essential roles in our implementation.