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arXiv:cond-mat/0502503 · Journal of Physics A Mathematical and General

Quantum logical states and operators for Josephson-like systems

Lara Faoro, Francesco A Raffa, Mario Rasetti

Published Jan 18, 2006• 4 citations
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Abstract

We give a formal algebraic description of Josephson-type quantum dynamical systems, i.e., Hamiltonian systems with a cos ϑ-like potential term. The two-boson Heisenberg algebra plays for such systems the role that the h (1) algebra does for the harmonic oscillator. A single Josephson junction is selected as a representative of Josephson systems. We construct both logical states (codewords) and logical (gate) operators in the superconductive regime. The codewords are the even and odd coherent states of the two-boson algebra: they are shift-resistant and robust, due to squeezing. The logical operators acting on the qubit codewords are expressed in terms of operators in the enveloping of the two-boson algebra. Such a scheme appears to be relevant for quantum information applications.

DOI
10.1088/0305-4470/39/5/l01

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We give a formal algebraic description of Josephson-type quantum dynamical systems, i.e., Hamiltonian systems with a cos ϑ-like potential term. The two-boson Heisenberg algebra plays for such systems the role that the h (1) algebra does for the harmonic oscillator. A single Josephson junction is selected as a representative of Josephson systems. We construct both logical states (codewords) and logical (gate) operators in the superconductive regime. The codewords are the even and odd coherent states of the two-boson algebra: they are shift-resistant and robust, due to squeezing. The logical operators acting on the qubit codewords are expressed in terms of operators in the enveloping of the two-boson algebra. Such a scheme appears to be relevant for quantum information applications.
We give a formal algebraic description of Josephson-type quantum dynamical systems, i.e., Hamiltonian systems with a cos -like potential term. The two-boson Heisenberg algebra plays for such systems the role that the h(1) algebra does for the harmonic oscillator. A single Josephson junction is selected as a representative of Josephson systems. We construct both logical states (codewords) and logical (gate) operators in the superconductive regime. The codewords are the even and odd coherent states of the two-boson algebra: they are shift-resistant and robust, due to squeezing. The logical operators acting on the qubit codewords are expressed in terms of operators in the enveloping of the two-boson algebra. Such a scheme appears to be relevant for quantum information applications
Our investigation concerns the class of Josephson-like systems, sharing the same nonlinear Hamiltonian. Among the latter a Josephson junction with an external biasing circuit is considered. We diagonalize the fully nonlinear Hamiltonian (in the superconductive regime of the junction) in the Fock space of the TBHA (two-boson Heisenberg algebra) and prove that such algebra leads quite naturally to the theoretical realization of codewords and logical operators: the codewords are defined as the even and odd coherent states of the TBHA, while the logical operators are expressed in terms of operators in the same algebra. Our theoretical construction corresponds to a continuous variable quantum computation scheme; the continuous variables are identified in terms of the physical operators of the junction. The link between this scheme and the technique of fermionization of bosonic systems is also discussed.
Any quantum computational network can be constructed with a sequence of the two-qubit diagonal quantum gates and one-qubit gates in two-state quantum systems. The universal construction of these quantum gates in the quantum systems and of the quantum computational networks with these gates may be achieved with the help of the operator algebra structure of Hamiltonians of the systems and the properties of the multiple-quantum operator algebra subspaces of the Liouville operator space and the specific properties of the quantum algorithm corresponding to the quantum network. As an example, the two-qubit diagonal gates are exactly prepared in detail in superconducting Josephson junctions.
We consider the superconductive regime of a Josephson junction with an external biasing circuit. After recalling that the eigenvalue equation of the junction Hamiltonian HJ is a special case of the Mathieu equation, we diagonalize HJ in a suitable Fock space and prove that this leads quite naturally from the use of the two‐boson algebra. We utilize then the normalized even and odd coherent states of the two‐boson algebra to investigate the squeezing properties inherent in the nonlinearity of the junction with respect to the Cooper pair number and phase operators, N̂ and ϑ̂. This task is accomplished by utilizing the relationship between N̂, ϑ̂ and the generators of the dynamical quantum algebra of the junction.
Recent experiments motivated by applications to quantum information processing are probing a new and fascinating regime of electrical engineering—that of quantum electrical circuits—where macroscopic collective variables such as polarization charge and electric current exhibit quantum coherence. Here I discuss the problem of constructing a quantum mechanical Hamiltonian for the low-frequency modes of such a circuit, focusing on the case of a superconducting qubit coupled to a harmonic oscillator or resonator, an architecture that is being pursued by several experimental groups. 1 Quantum gate design In the quantum circuit model of quantum information processing, an arbitrary unitary transformation on N qubits can be decomposed into a sequence of certain universal two-qubit logical operations acting on pairs of qubits, combined with arbitrary single-qubit rotations [1]. The purpose of quantum gate design is to develop experimental protocols or “machine language code ” to implement these elementary operations. For quantum information processing architectures based on superconducting circuits [2, 3], the first step is to construct an effective Hamiltonian for the system. Whereas the fully