Coherent controlization using superconducting qubits

Coherent controlization, i.e., coherent conditioning of arbitrary single- or multi-qubit operations on the state of one or more control qubits, is an important ingredient for the flexible implementation of many algorithms in quantum computation. This is of particular significance when certain subroutines are changing over time or when they are frequently modified, such as in decision-making algorithms for learning agents. We propose a scheme to realize coherent controlization for any number of superconducting qubits coupled to a microwave resonator. For two and three qubits, we present an explicit construction that is of high relevance for quantum learning agents. We demonstrate the feasibility of our proposal, taking into account loss, dephasing, and the cavity self-Kerr effect.

A superconducting LC-circuit may be thought of as a harmonic oscillator, where the position and conjugate momentum variables are the flux Φ through the inductor and the charge Q on the capacitor plates, respectively. Indeed, these quantities have to be treated as operators satisfying the canonical commutation relation [ Φ , Q ] = i , see, e.g., Ref. [1]. In other words, the system Hamiltonian H = Q 2 /(2C) + Φ 2 /(2L) may be written as H = ω(q † q + 1/2) (A.1) in terms of the ladder operators defined by with ω = 1/ √ LC, Z = L/C, and q, q † = 1. The introduction of a Josephson junction (a thin insulating barrier separating two pieces of superconducting material) creates a non-linearity in the system that allows * nicolai.friis@uibk.ac.at identifying two energy levels as the qubit levels. The Hamiltonian is then modified to where the junction inductance L J is typically expressed via the Josephson energy E J , that is, L J = Φ 2 0 /E J , with the magnetic flux quantum Φ 0 = /(2e), and Φ/Φ 0 is the phase difference across the junction. The original capacitance and the Josephson junction now form a superconducting island -a Cooper-pair box (CPB) -with capacitance C Σ = C + C J , which can be written in terms of the charging energy E C = e 2 /(2C Σ ). The charge in the CPB depends on n, the number of transferred Cooper pairs (with charge 2e), and the effective offset charge 2en o , i.e., Q/(2e) = n−n o . With the replacement Z → Z J = L J /C Σ one may define raising and lowering operators in full analogy to (A.2). To remove the strong dependence of the energy-level splitting on the (environmentally induced) offset charge, an additional shunting capacitor with large capacitance C B can be inserted into the circuit in parallel with the Josephson junction to create the transmon qubit [2]. This strongly increases C Σ → C Σ = C B + C + C J , such that E C /E J 1. Inserting the ladder operators q and q † (with Z → Z J ) into Eq. (A.3) and expanding in powers of E C /E J we arrive at Within the rotating wave approximation, the Hamiltonian of Eq. (A.4) can be truncated to the effective transmon qubit Hamiltonian where ω Q ≡ √ 8E J E C − E C / is the transmon qubit frequency and β ≡ E C / is the anharmonicity.
A.II. Transmons coupled to a resonator Next, we are interested in describing a transmon qubit that is capacitively coupled to a microwave resonator. Within the rotating wave approximation the joint system is described by the Hamiltonian where g is the qubit-cavity coupling, ω R and ω Q are the frequencies of the isolated qubit and cavity, respectively, and the operators c and c † are the resonator ladder operators satisfying [ c , c † ] = 1. The terms in H qr that are quadratic in the mode operators can be diagonalized by a Bogoliubov transformation. That is, one introduces the dressed operators a and b, such that with tan 2θ = 2g/∆, where ∆ ≡ ω Q − ω R is the qubitcavity detuning. The excitations of the dressed modes can no longer be uniquely attributed to just the resonator or just the qubit. However, in the strong dispersive regime, where g/∆ 1, the mixing angle is small, θ ≈ g/∆ and excitations created by c † (q † ) are associated "mostly" with the cavity (qubit). The Hamiltonian transforms to where the dressed mode frequencies are given bỹ The terms in the square bracket can be seen to oscillate rapidly, and we may therefore remove these terms in another rotating wave approximation. With the notation χ qr ≡ (β/2) sin 2 (2θ), χ qq ≡ β cos 4 θ, and we then arrive at the Hamiltonian of a single transmon qubit coupled to a resonator in the dispersive limit where ω r =ω r +χ rr /2 and ω q =ω q +χ qq /2. The dressed qubit anharmonicity χ qq , the qubit-cavity cross-Kerr coefficient χ qr , and the cavity self-Kerr coefficient χ rr can be expressed directly via the coupling strength g and the detuning ∆ as A.III. Two transmons coupled to a cavity When two transmon qubits are coupled to the cavity, we may write an analogue expression to Eq. (A.6). That is, in the rotating wave approximation we have the Hamiltonian where we have neglected any direct coupling of the qubits to each other. In the two-qubit case (and beyond), an analytical diagonalization of the harmonic part (terms quadratic in the mode operators) becomes infeasible. However, using numerical methods and following similar arguments as presented in the previous Sec. A.II one arrives at the effective Hamiltonian (A.14) Assuming that the transmons can be fabricated such that χ qiqi ≡ χ qq for all qubits, we have numerically checked that the cavity self-Kerr coefficient χ rr can be approximated as

B. SIMULATIONS OF COHERENT CONTROLIZATION USING TRANSMON QUBITS
In this section we present the simulations of our protocol for coherent controlization with two and three qubits. In the simulations, which were coded in PYTHON using the QuTiP library [3], we discuss the influence of the cavity Kerr effect, including the (partial) correction of the linear Kerr effect by way of reference frame adjustments, and the possibility for correcting it entirely using photonnumber selective gates [4]. In addition, all simulations assume the presence of amplitude and phase damping for the qubits, and photon loss in the resonator. The unconditional displacements can safely be assumed to be perfect. We further assume the single-qubit operations to be perfect, given that the pulses are slow enough to address only the zero-photon subspace, but short enough to fit within the ∆t time intervals of our protocol.

B.I. Setup for simulations
For the simulation we truncate the Hilbert space of the resonator to be spanned by the Fock states | n r of photon numbers n = 0, 1, . . . , 100. This is a good approximation since the maximal average photon number of the coherent states in our protocol isn max = |2α| 2 = 14 (n max = |4α| 2 = 32) photons for 2 (3) qubits and the overlap with photon numbers larger than 100 is hence negligible. For the Hilbert space of the transmon qubits we will each only consider the lowest two eigenstates, i.e., the qubit levels. The effective Hamiltonian in the frame rotating with ω r and ω q for the resonator and qubit Hilbert spaces, respectively, is given by where σ ± j = σ x j ± iσ y j are the raising/lowering operators of the j-th qubit. In addition to the free time evolution we will include corrections for the linear order of the Kerr effect by including rotations U ϕ (γ) = exp(−iϕ γ a † a) after each period of time evolution, where γ is the maximal displacement of the different coherent state components in the preceding step of the protocol. The conditional single-qubit operations will be represented by where the single-qubit Y -rotations on the second (and third) of the two (three) qubits are realized by a timedependent drive U (θ j ) = exp(−i θj 2 σ y t T ). The durations T of these drives are taken to be the durations of the waiting periods in the corresponding protocol steps (T = ∆t for two qubits). On top of the unitary time evolution, unconditional displacements, and conditional qubit operations, all of which will be assumed to be perfect, we will consider decoherence in the system. In particular, we assume that the dynamics of the overall state ρ of the joint cavity-qubit system during the waiting periods of the protocol is governed by the master equation As the output of interest of the simulations we consider the fidelities F r for the resonator, i.e., the squared overlap with the vacuum at the final step of the protocol, given by F r = r 0 | Tr q ρ | 0 r , and F q for the qubits, i.e., the squared overlap with the target state | ψ q , given by F q = q ψ | Tr r ρ | ψ q . We evaluate these two fidelities separately because the success of the protocol is ultimately only determined by achieving a large overlap of the reduced qubits state with the desired target state, irrespective of the final cavity state. On the other hand, the cavity fidelity gives a more detailed overview of the errors incurred by the Kerr effect in the resonator. Therefore, both F r and F q are quantities of interest for our protocol. In addition to the final fidelities, we include plots of the Wigner function W (x, y), where of the reduced resonator state ρ r = Tr q ρ, where | x are the eigenstates of the quadrature operator Φ with eigenvalue x, and plots of the absolute values of the density matrix elements | q µν| ρ q |mn q | of the reduced qubit state ρ q = Tr r ρ throughout the steps of our protocol.

B.II. Simulations for two qubits
For two qubits, the qubit-cavity cross-Kerr coefficients χ qr ≡ χ q1r = 2χ q2r are chosen in a range from 1.5 × 2π MHz to 3 × 2π MHz. According to the approximation in Eq. (A.15), we further set χ rr = 5χ 2 qr /(4χ qq ), where χ qq is fixed to 300 × 2π MHz. The coherence time for the resonator is set to τ r = 100 µs throughout. For the simulation results presented in Fig. B.1, we hence vary the qubit-cavity coupling, the displacements in the protocol, the decoherence times, the initial qubit state ρ q = | ξ ξ | q and the angles θ j (j = 1, 2, 3) for the singlequbit operations.
In addition to the results shown in Fig. B.1, we have simulated the influence of decoherence on the protocol when assuming that the cavity Kerr effect has been fully corrected using the methods of [4]. In that case, using the same parameters as for the simulation shown in Fig. 2 (c,d) of the main text, but setting χ rr ≡ 0, we obtain F r = 99% and F q = 96%.

B.III. Simulations for three qubits
For three qubits, the qubit-cavity cross-Kerr coefficients χ qr ≡ χ q1r = 2χ q2r = 4χ q3r are chosen in a range from 0.2 × 2π MHz to 0.4 × 2π MHz. According to the approximation in Eq. (A.15), we further set χ rr = 21χ 2 qr /(4χ qq ), where χ qq is fixed to 300 × 2π MHz. The coherence times for the resonator and the qubit (dephasing and amplitude damping) are set to 100 µs. For the simulation results presented in Table B.1, we hence vary the qubit-cavity coupling, the displacements in the protocol, the initial qubit state ρ q = | ξ ξ | q and the angles θ j (j = 1, 2, . . . , 7) for the single-qubit operations. A selection of these simulations Figs. B.3 and B.4 further illustrate the reduced states of the resonator and the qubits throughout the protocol.