Analysis of a parametrically driven exchange-type gate and a two-photon excitation gate between superconducting qubits

Abstract

A current bottleneck for quantum computation is the realization of high-fidelity two-qubit quantum operations between two and more quantum bits in arrays of coupled qubits. Gates based on parametrically driven tunable couplers offer a convenient method to entangle multiple qubits by selectively activating different interaction terms in the effective Hamiltonian. Here, we study theoretically and experimentally a superconducting qubit setup with two transmon qubits connected via a capacitively coupled tunable bus. We develop a time-dependent Schrieffer-Wolff transformation and derive analytic expressions for exchange-interaction gates swapping excitations between the qubits (iSWAP) and for two-photon gates creating and annihilating simultaneous two-qubit excitations (bSWAP). We find that the bSWAP gate is generally slower than the more commonly used iSWAP gate, but features favorable scalability properties with less severe frequency crowding effects, which typically degrade the fidelity in multi-qubit setups. Our theoretical results are backed by experimental measurements as well as exact numerical simulations including the effects of higher transmon levels and dissipation.Quantum computation is based on accurate and precise control of quantum bits and their interactions to create multi-qubit superpositions and entanglement. With superconducting circuits, single qubit quantum gates can be carried out with fidelities approaching 99.99% [1-4], while errors in two-qubit operations are typically higher with record fidelities around 99% [3,5]. However, the realization of qubit operations with even higher fidelity is required both for reaching the error threshold for quantum computation [6-9] and for carrying out reliable quantum simulations and optimizations in large arrays of coupled qubits [10][11][12][13]. Moreover, the quest for useful quantum computations before full quantum error correction becomes available may be assisted by efficient, short-depth gate sequences based on two-or multi-qubit gates [14,15] with versatile types of interactions. In particular, parametric schemes based on tunable couplers have been proposed and recently realized as a means to achieve fast gates with high fidelities [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].In this context, effective interactions were engineered in Ref.[26] between two transmon qubits mediated by a third, ancilla transmon device (bus), which couples dispersively to both qubits and whose frequency is modulated by an external magnetic flux. Such a flux-modulation scheme provides frequency-selectivity and allows to use fixed-frequency computational qubits, thereby minimizing the sensitvity of the device with respect to magnetic flux noise and disorder effects. For example, modulating at the (fixed) difference frequency of the qubits brings these qubits effectively into resonance in a co-rotating frame such that a single excitation can be swapped efficiently (iSWAP). The effective Hamiltonian in this case is H iSWAP ∝ XX + YY. With this method gate ...

Gate-Efficient Simulation of Molecular Eigenstates on a Quantum Computer

“…An exchange-type gate primitive can naturally be realized in a tunable coupler architecture ( Fig. 2) [29,30]. The device consists of two fixed-frequency transmon qubits Q1 and Q2 linked via a tunable coupler (TC), i.e.…”

“…The master equation (Eq. 7) withĤ tr representing the coupled system of two transmons and a tunable coupler, implemented as three level systems, is numerically solved using QuTiP [30,43]. From this solution, the density matrix of the evolved state is calculated as a function of θ and φ.…”

“…Spectroscopic measurements of the device yield qubit frequencies ω 1,2 , capacative coupling strengths g 1,2 and decay rates as summarized in Table I. An exchange-type coupling between the computational qubits Q1 and Q2 is achieved by parametric modulation of the TC frequency ω c (t) = ω 0 c |cos(πΦ(t)/Φ 0 )| [29,30]. Threading a magnetic fluxthe SQUID loop of the TC implements the effective Hamiltonian [30]with the set of Pauli operators {X, Y, Z} ≡ {σ x ,σ y ,σ z }.…”

“…We demonstrate in simulation that the circuit depth required to achieve arXiv:1809.05057v1 [quant-ph] 13 Sep 2018 chemical accuracy in a VQE algorithms can be significantly reduced by using exchange-type gates, which would allow the simulation of larger quantum systems on near-term quantum hardware. We implement such an exchange-type gate based VQE algorithm on a hardware platform consisting of two fixed-frequency superconducting qubits coupled via a tunable coupler [29,30] and determine the ground state energy of molecular hydrogen. Finally, we derive the excited states of molecular hydrogen from the measured ground state using the equation-of-motion (EOM) approach [31], which complements the quantum subspace expansion in [25,32].…”

A quantum engineer's guide to superconducting qubits

“…This phase can be conveniently removed either by subsequent application of virtual-Z gates to all following pulses 191 , or by shaping the waveform of the excursion such that single-qubit phases are exactly cancelled 222 . Equations (104) and (108) together present a useful result from a quantum processor design perspective: The operating regime, frequency and time τ of the iSWAP can be calculated (typically simulated) to high precision, before any processor fabrication is undertaken. The only 'quantum' parts that enter g qq (and g q-r-q ) are the qubit frequencies, ω q1 (Φ 1 ) and ω q2 (Φ 2 ).…”