Effective Hamiltonians for Interacting Superconducting Qubits--Local Basis

Effective Hamiltonians for interacting superconducting qubits: local basis reduction and the Schrieffer-Wolff transformation Gioele Consaniy and Paul A. Warburtony yUniversity College London March 27, 2020 Abstract An open question in designing superconducting quantum circuits is how best to reduce the full circuit Hamil- tonian which describes their dynamics to an effective two-level qubit Hamiltonian which is appropriate for ma- nipulation of quantum information. Despite advances in numerical methods to simulate the spectral properties of multi-element superconducting circuits[1, 2, 3], the literature lacks a consistent and effective method of deter- mining the effective qubit Hamiltonian. Here we address this problem by introducing a novel local basis reduction method. This method does not require any ad hoc assumption on the structure of the Hamiltonian such as its linear response to applied fields. We numerically benchmark the local basis reduction method against other Hamiltonian reduction methods in the literature and report specific examples of superconducting qubits, including the capacitively-shunted flux qubit, where the standard reduction approaches fail. By combining the local basis reduction method with the Schrieffer-Wolff transformation we further extend its applicability to systems of interacting qubits and use it to extract both non-stoquastic two-qubit Hamiltonians and three-local interaction terms in three-qubit Hamiltonians. 1 Introduction acterised by the fact that, under specific operation con- ditions, they can be regarded as two-level systems (in the Since their first appearance, superconducting (SC) cir- sense that any additional stationary state of the system cuits including Josephson junctions have proved to be has a substantially higher energy and a small probability one of the most promising platforms for quantum in- of being populated)[16]. formation processing applications[4, 5, 6, 7, 8]. The The fundamental theory describing SC circuits, i.e. lithographic fabrication process allows fine tuning of the quantum network theory, is well established and can be physical properties of each superconducting circuit, thus used, at least in some approximate form, to numerically resulting in qubits with different spectral properties. In- determine the energy spectrum of an arbitrary SC qubit dividual qubits can be manufactured in large arrays, circuit[1, 2, 3]. The literature seems, however, to be with electrostatic and magnetic interactions coupling missing an agreed and consistent way of connecting the pairs of them. The strength of the local fields on each electromagnetic Hamiltonian Hê:m: of an arbitrary sys- qubit and of the two-qubit interactions can further be tem of n SC qubits to the corresponding effective qubit adjusted dynamically by applying external electrostatic Hamiltonian H^1, or, equivalently, of numerically deter- arXiv:1912.00464v2 [quant-ph] 25 Mar 2020 and magnetic fields, making for a flexible and scalable mining the parameters of an n-spin Hamiltonian which architecture for both gate-based quantum computation reproduces the low-energy spectrum of Hê:m:, as well as (GBQC) and quantum annealing (QA)[9, 5, 4, 10, 11]. the computational state probabilities and the expecta- A two decades quest to improve the coherence met- tion values of the system observables. As we will see rics of superconducting qubits, by materials and circuit below, where such mapping methods do exist (see, for engineering, has led to a number of SC qubit designs, instance, supplementary materials of Ref. [17, 18]), they such as capacitively-shunted flux qubits and transmons, are not guaranteed to reproduce the correct low-energy having T1 and T2 times in the 100 μs range[12, 13]. spectrum of the circuit. These circuits, as much as the earlier designs, includ- A general scheme for reducing the circuit Hamilto- ing rf-SQUID qubits[14], persistent-current qubits[9] and nian of an arbitrary SC qubit system to the correct ef- single-Cooper-pair boxes[15] are, by construction, char- fective qubit Hamiltonian would serve several purposes. 1 Firstly, the effective qubit Hamiltonian can be used to scribing the superconducting circuit, we begin this paper model and interpret quantum state evolution experi- by revising how to write down the circuit Hamiltonian for ments, since it contains all the necessary information to a generic non-dissipative circuit. We start with isolated describe the dynamical evolution of the qubit system, as circuits and later consider the presence of interactions. long as this is not excited outside of the computational The framework which we use is that of quantum network space (leakage)[7, 18], while being much more compact theory, which is the quantum version of Lagrangian me- than the full circuit Hamiltonian. Secondly, specifically chanics applied to electrical circuits[1, 24]. Following in the context of adiabatic quantum computing (AQC), the standard procedure we will first write the classical identification of non-stoquastic and multi-local terms in Hamiltonian and then quantise it by replacing the vari- the qubit Hamiltonian could help the engineering of ables with the corresponding operators. The reader who such terms, which are fundamental to implement non- is familiar with these concepts may wish to skip to the stoquastic AQC (which is thought to be more powerful next section. than its stoquastic counterpart[19]) and error suppres- sion protocols based on stabiliser codes[20], respectively. 2.1 Isolated circuits In experiments involving such non-stoquastic and multi- body interaction terms, the analysis of spectroscopic The first key assumption we make in order to apply data can also be made substantially easier by the avail- quantum network theory is that the size of our qubit ability of the reduced Hamiltonian[18]. Lastly, in the is sufficiently small relative to microwave wavelengths, case of single qubits, the calculation of the effective qubit such that it is appropriate to use a lumped-element de- Hamiltonian represents an improved way of estimating scription of the circuit[25]. Because the system is super- the tunnelling amplitudes between semi-classical poten- conductive, this circuit will consist of nodes connected tial minima that is an alternative to instanton-based ap- by branches containing only non-dissipative elements, proaches and therefore potentially more accurate, espe- namely inductors, capacitors and Josephson junctions. cially in the limit of large tunnelling amplitudes[21, 22]. An example representing the equivalent circuit of an rf- In this paper we propose a method of implementing SQUID flux qubit is shown in figure 1. Then, without Hamiltonian reduction based on a natural local defini- loss of generality, we can arbitrarily assign one of the tion of the computational basis. Our method does not circuit nodes to ground. (For a floating qubit there will require any ad hoc assumption on the structure of the be a capacitor between the ground node and the rest of Hamiltonian, such as its linear response to the applied the circuit.) electrostatic and magnetic fields, which is at the core At this point, in order to later take into account the of standard perturbative reduction methods[17]. Ad- effect of external magnetic fields, we need to choose a ditionally the scheme can be applied to individual SC spanning tree, i.e. a path of connected branches going qubits of any kind, as well as to systems of qubits and from the ground node to every other node, without gen- coupler circuits, interacting magnetically or electrostat- erating loops. The specific choice of the spanning tree ically. In the interacting case the scheme makes use of will not affect our final results[24]. A possible choice of the Schrieffer-Wolff transformation to separate the low the spanning tree for the rf-SQUID flux qubit in figure 1 energy subspace from the rest of the Hilbert space[23]. The article is structured as follows: in the next section we revise how to write a general electromagnetic Hamil- tonian for isolated and coupled superconducting circuits. In section 3 we introduce some of the state-of-the-art reduction methods in the literature and then present our novel approach to the problem, in the context of both single and interacting qubits. In section 4 we present the numerical results of Hamiltonian reduction applied to systems of superconducting qubits, with a specific ref- erence to recent publications. Finally we summarise our conclusions. 2 Circuit Hamiltonians from Quantum Circuit Analysis Figure 1: Equivalent lumped-element circuit of an rf- Since we want to establish a way to numerically derive an SQUID flux qubit[14]. One of the two possible choices effective qubit Hamiltonian from the full Hamiltonian de- of the spanning tree is highlighted in red. 2 is highlighted in red. We will indicate the set of branches branch bij connecting nodes i and j (the index 0 refers in the spanning tree by T and the complementary set of to the ground node here) and Φ0 = h=(2e) ' 2:0678 · closure branches by C. Every closure branch is associ- 10−15Wb is the magnetic flux quantum. The branch ated with an irreducible loop in the circuit, which is the fluxes fΦbij gj>i=0;:::;N appearing inside the expression smallest loop formed by that closure branch and by other are defined as branches in the spanning tree. For instance, in the flux ( qubit in Fig. 1 the closure branch b is associated with Φ − Φ ; if b 2 T ; 01 Φ = i j ij (7) the single loop in the circuit[24]. bij ext Φi − Φj + Φij ; if bij 2 C; Every state of our circuit is defined by specifying the instantaneous voltages at each of the nodes. Al- where Φext is the external magnetic flux threading the ternatively, we can define, for every node j (excluding ij irreducible loop associated with bij. ground), a node flux variable Φj, representing the inte- Our definition of the branch fluxes includes the effect gral over time of its voltage, i.e.

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