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Quantum computing with electrical circuits : Hamiltonian construction for basic qubit-resonator models
| Content Provider | Semantic Scholar |
|---|---|
| Author | Geller, Michael |
| Copyright Year | 2007 |
| Abstract | 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 microscopic Hamiltonian for the electronic and ionic degrees of freedom in the conductors forming the circuit is known, at least in principle, the Hamiltonian of interest here describes only the relevant low-energy modes of that circuit. A rigorous construction might involve making a canonical transformation from the microscopic quantum degrees of freedom to a set of collective modes. Here I follow a simpler and more intuitive phenomenological quantization method, whereby a classical description based on Kirkoff’s laws is derived first, and then later canonically quantized. It is important to realize that such an approach is not based on first principles and must be confirmed experimentally. 2 The phase qubit The primitive building block for any superconducting qubit is the Josephson junction (JJ) shown in Fig. 1. The low-energy dynamics of this system is governed by the phase difference φ between the condensate wave functions or order parameters on the two sides of the insulating barrier. The phase difference is an operator canonically conjugate to the Cooper-pair number difference N , according to [φ,N ] = i. (1) The low-energy eigenstates ψm(φ) of the JJ can be regarded as probability-amplitude distributions in φ. As will be explained below, the potential energy U(φ) of the JJ is manipulated by applying a bias current I to the junction, providing an external control of the quantum We define the momentum P to be canonically conjugate to φ, and N ≡ P/h̄. In the phase representation, N = −i ∂ ∂φ . |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.physast.uga.edu/~mgeller/59.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Architecture as Topic Biasing Canonical quantization Coherence (physics) Computation (action) Electrical engineering Electrical junction Electricity Esophagogastric Junction Experiment HL7PublishingSubSection |
| Content Type | Text |
| Resource Type | Article |
