nSQUID ARRAYS AS CONVEYERS OF QUANTUM INFORMATION
Qiang Deng, Dmitri V. Averin*
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794-3800
Received August 4, 2014
We have considered the quantum dynamics of an array of nSQUIDs — two-junction SQUIDs with negative mutual inductance between their two arms. Effective dual-rail structure of the array creates additional internal degree of freedom for the fluxons in the array, which can be used to encode and transport quantum information. Physically, this degree of freedom is represented by electromagnetic excitations localized on the fluxon. We have calculated the spatial profile and frequency spectrum of these excitations. Their dynamics can be reduced to two quantum states, so that each fluxon moving through the array carries with it a qubit of information. Coherence properties of such a propagating qubit in the nSQUID array are characterized by the dynamic suppression of the low-frequency decoherence due to the motion-induced spreading of the noise spectral density to a larger frequency interval.
Contribution for the JETP special issue in honor of A. F. Andrew's 75th birthday
DOI: 10.7868/S0044451014120189
1. INTRODUCTION
Coherence properties and precision of control over the dynamics of superconducting qubits (see, e.g., recent experiments fl 5]) have reached the level when it becomes possible and interesting to discuss potential architecture of the superconducting quantum computing circuits either within the gate-model paradigm [6 9] or the adiabatic ground-state approach [10, 11]. Besides the formidable problem of maintaining the level of qubit coherence with increasing circuit complexity, the central issue that needs to be addressed by any architecture of scalable quantum computing devices is the requirement of rapid transfer of quantum information among a large number of qubits with sufficient fidelity. So far, the suggested solutions to the problem of transfer of quantum information were based on controllable direct coupling of qubits or coupling through a common resonator. These solutions, while working nicely for the circuits of few qubits, can not be scaled easily to larger circuits.
The purpose of this work is to suggest another approach to the problem of information transfer along a quantum circuit of the superconducting qubits utilizing quantum dynamics of magnetic flux, in which the
E-mail: dmitri.averin'&stonybrook.edu
quantum information is transported along the circuit by propagating classical pulses. This approach uses the arrays of two-junction SQUIDs, where each of them has a negative mutual inductance between its two arms. Dynamics of such "nSQUIDs" [12] can be represented in terms of the two degrees of freedom, the "differential" mode and the "common" modes, with very different properties. The former can be used to encode quantum information, while the latter to transport it. Then, the overall architecture of a quantum computing circuit built of nSQUIDs is very similar to superconducting classical reversible circuits also based on nSQUIDs [13, 14], in which the computation is organized around the information-carrying pulses propagating along the circuit.
Existence of the two degrees of freedom with different properties makes nSQUID arrays qualitatively and advantageously different from previously considered arrays of superconducting qubits (see, e.g., [15 17]) or spins [18 20] as tools for quantum information transfer. In general, a physical variable encoding quantum information and the one used to carry it should satisfy completely different sets of requirements, and in the case of nSQUIDs, the common and the differential dynamic modes can be optimized separately to satisfy these different requirements. Most importantly, the common mode, i.e. the degree of freedom transporting the quantum information, does not necessarily
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nood to bo itself quantum, since a classical dynamics is sufficient to transport a qubit along the array. This makes potential operation of nSQUID arrays as conveyers of quantum information considerably more straightforward, since it avoids a challenging problem of maintaining quantum coherence of fluxons, supported by the common mode, along a large array of junctions.
2. BASIC MODEL OF nSQUID ARRAY
We begin our detailed discussion with a description of the elementary cell of the arrays considered in this work, i. o., nSQUID [12] a two-junction SQUID with a negative inductance between its two arms (Fig. 1). Dynamics of this structure can be separated naturally into the dynamics of two degrees of freedom, i.e., the common mode representing the total current flowing through the two junctions of the SQUID and the differential mode which represents the difference of the two junction currents, i. o., the current circulating along the SQUID loop. In the configuration of an one-dimensional array, these two degrees of freedom give rise to the two different excitation modes of the array. The common mode corresponds to excitations propagating along the array and, in the appropriate regime, takes the form of individual fluxons.
The main difference between the nSQUID and the usual two-junction SQUID can be seen if one thinks very crudely about two arms of a SQUID as two parallel wires. For the plain wires, the mutual inductance M between the wires is positive, M > 0, ensuring that the effective inductance of the differential mode is always smaller than that of the common mode. In this case, the differential mode can have a non-trivial dynamics only together with the common mode. By contrast, the negative mutual inductance —M between the SQUID arms makes the effective inductance of the differential mode larger than the inductance of the common mode. As a result, one can realize a situation when the differential mode exhibits a non-trivial, e. g., bi-stablo dynamics at low frequencies without exciting the common mode which supports only the excitations with larger frequencies. The differential mode can then be used to encode information, while the dynamics of the common mode is optimized separately for transfer of this information along the array. If the dynamics of both modes is classical, nSQUID structures provide a convenient basis for implementation of classical reversible computing [13, 14]. If, however, parameters of the differential mode are such that its behavior is quantum, it can be used to encode quantum information, which can
Fig. 1. Equivalent circuit of an nSQUID, i.e., a two-junction SQUID with junction capacitances C and Josephson coupling energies Ej and the negative mutual inductance —M between its inductive arms with inductances L. The negative mutual inductance makes the effective inductance of the common mode of the SQUID dynamics much smaller than the inductance of the differential mode. Also shown are the phase bias of the common mode and 6,-: of the differential mode
then be transported along the array by the evolution of the common mode.
Hamiltonian H of the individual nSQUID (Fig. 1) is given by the standard expression which includes the charging energies and the Josephson coupling energies of the two SQUID junctions. Adding the magnetic energy of the two inductive arms of the SQUID coupled by the negative mutual inductance, we obtain the following expression for the nSQUID Hamiltonian:
H =
K 2 Cf,
91
4 C
2 7T
'11'..I COS \ COS <j) ■ (:v-:Yc)2 ,
L - M
L + M
(1)
Here = ttH/c is the magnetic flux quantum, K and \ are the variables of the common mode: K = Qi + Q-2 is the total charge on the two capacitances of the SQUID junctions, and \ = (<j>i + 02)/2 is the average Joseph-son phase difference across the junctions. The common mode has effective inductance (L — M)/2 and capacitance Ci, which, in the case of the circuit in Fig. 1, is equal to the total capacitance 2C of the two junctions, but in general can include additional contribu-
Fig. 2. Dual-rail Josephson array made of nSQUID cells shown in Fig. 1. The array cells are coupled by inductances Lc with negative mutual inductance —Mc between them. No bias phases are applied externally either for the common mode or the differential mode; the bias for the common mode is generated self-consistently by the array dynamics. In this dynamics, the common mode plays the role of the qubit control signal propagating along the control line with specific capacitance Co and inductance Lu, whereas the differential mode encodes a qubit of quantum information that is being transported along the array
tions from the external biasing circuit (as, e. g., in Fig. 2 below). In quantum dynamics, K and \ are canon-ically conjugated variables that satisfy the commutation relation A'] = 2ei, standard for the charge and phase of a Josephson junction. The corresponding variables of the differential mode are the charge difference Q = Qi — Q-2 and the phase difference <f> = (<f>i — <i>-2)/2 which have the same commutation relation [(f), Q] = 2ei. The effective inductance and capacitance of this mode are 2(L + M) and C/2 respectively. (Their apparent values in Eq. (1) are different because of the chosen normalization of Q and <f>.)
As mentioned above, qualitative effect of the negative mutual inductance is to make the dynamic properties of the common and the differential modes in the Hamiltonian (1) of one nSQUID very different. If we neglect for a moment the Josephson coupling, the resonance frequencies of the two modes are [2/(L-M)Ci]1^2 and 1/[(L + M)C}^2 = ujp, and in the large-negative-inductance limit M —¥ L, the excitation frequency of the common mode becomes much larger than that of the differential mode, making it possible to clearly separate the frequency ranges of the dynamics of the two modes. As a result, when the nSQUID cells are connected in an array as in Fig. 2, the two modes can be used to perform different functions. In partic-
ular, if the coupling inductances Lc are designed to have negative mu
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