A Linear Quantum Coupler for Clean Bosonic Control

TL;DR

The paper tackles the challenge of achieving fast, high-fidelity parametric control in superconducting circuits without triggering parasitic nonlinearities or decoherence. It introduces the Linear Inductive Coupler (LINC), a balanced, flux-biased, Kerr-free three-wave mixer whose idle state is truly linear at $\phi_{DC}=\pi/2$ and which enforces a parity-protection selection rule under drive to suppress unwanted processes. Analytical results yield simple expressions for the Kerr and three-wave-mixing strength, while Floquet analyses demonstrate that LINC offers cleaner driven spectra and higher fidelity than Kerr-full alternatives like the SNAIL, especially in multi-tone scenarios; arraying multiple LINCs can suppress driven Kerr by $1/M^2$, enabling scalable, high-fidelity bosonic and qubit control. The proposed approach promises significant impact on high-Q control, readout, and amplification, with practical path to integration in existing superconducting platforms and robust performance under realistic asymmetries and parasitics.

Abstract

Quantum computing with superconducting circuits relies on high-fidelity driven nonlinear processes. An ideal quantum nonlinearity would selectively activate desired coherent processes at high strength, without activating parasitic mixing products or introducing additional decoherence. The wide bandwidth of the Josephson nonlinearity makes this difficult, with undesired drive-induced transitions and decoherence limiting qubit readout, gates, couplers, and amplifiers. Significant strides have been recently made into building better `quantum mixers', with promise being shown by Kerr-free three-wave mixers that suppress driven frequency shifts, and balanced quantum mixers that explicitly forbid a significant fraction of parasitic processes. We propose a novel mixer that combines both these strengths, with engineered selection rules that make it essentially linear (not just Kerr-free) when idle, and activate clean parametric processes even when driven at high strength. Further, its ideal Hamiltonian is simple to analyze analytically, and we show that this ideal behavior is first-order insensitive to dominant experimental imperfections. We expect this mixer to allow significant advances in high-Q control, readout, and amplification.

Figures (9)

• Figure 1: The LINC, a quantum single-balanced mixera, Fundamentals of mixer balancing. By taking two identical nonlinear elements and driving them $180^\circ$ out of phase (e.g., with a balun), we can create a single-balanced mixer. If the nonlinear element is a single junction, this circuit is the familiar differentially-driven SQUID (DDS, SQUIDBeamsplitterPaper). If the nonlinear element is instead a DC flux-biased RF-SQUID, this circuit results in the Linear Inductive Coupler (LINC). The $\Pi$-T transformation provides a simpler circuit representation. b, An illustration of the advantage of a balanced mixer. In general, a driven nonlinear element will permit numerous nonlinear processes, of which only a small subset is typically desired. Balancing the nonlinear element will suppress a significant fraction of spurious processes (light orange), but preserve the desired process (black) at high strength. Any remaining parasitic processes (red) still permitted by the symmetry must be avoided through other means, like careful selection of drive frequencies. c, The LINC circuit, with a symmetric loop and capacitive pads, enables a separation of driven and undriven mixer behavior. At the operating point where the total DC flux in the outer SQUID loop is half a flux quantum, the loop junctions are biased to effectively infinite inductance, leaving only the linear shunt as the coupler's inductor. This simultaneously nulls all nonlinearity when idle, which is beyond just being Kerr-free. When threaded with AC flux, the outer loop activates and drives a balanced three-wave mixing process.
• Figure 2: Static Hamiltonian and flux noise sensitivity.a, The static LINC frequency and Kerr as a function of DC flux. Numerical values (solid lines) are calculated by exact diagonalization of a LINC Hamiltonian with $E_L/h=52.8$ GHz, $E_J/h=15.84$ GHz, $E_C/h=100$ MHz, with the center shunt composed of an array of 10 junctions, where $h$ is Planck's constant. Overlayed analytic curves follow \ref{['eq:analytic_Kerr', 'eq:analytic_freq']}. At the operating point of $\phi_{\text{DC}}=\pi/2$, the coupler is linear. b, Three-wave mixing strength (dashed black) and sensitivity to flux-noise (yellow) as a function of DC flux. The former is calculated from a Floquet simulation of a parametric squeezing process. The latter is a simple derivative of the static frequencies calculated in part a ($1/2\pi \;d\omega_L/d\Phi$). These quantities are equal, up to a scaling factor of $2$ (see \ref{['app:Circuit']}). c, Inherited dephasing for a coupled quantum memory, as a function of DC flux. Thermal noise-induced dephasing is calculated for $n_{\text{th}}^{\text{coupler}} = 2\%, T_1 = 20 \mu$s, and is minimized at the operating point, where the coupler is linear with no dispersive shift $\chi$ to the resonator. Arraying multiple LINCs together dilutes nonlinearity and therefore $\chi$, broadening the range over which $\chi \lesssim 1/T_1$ and dephasing is suppressed. The low-frequency dephasing due to inherited flux noise is calculated through \ref{['eq:kappa_phi_cav']}, with noise amplitude $A_\Phi = 1\mu\Phi_0/\sqrt{Hz}$.
• Figure 3: The LINC as a driven mixera, LINC beamsplitting strength ($g_{BS}$) and Kerr ($\alpha_{L}$) as a function of drive strength ($\phi_{\text{AC}}$), from an exact Floquet simulation. While the LINC is Kerr-free when idle, higher-order nonlinearities can induce a driven Kerr. This can be suppressed by arraying multiple LINCs while preserving the (driven) flux through each LINC loop (arrayed 3x, dashed). b, Driven LINC frequency shift (solid purple) vs drive strength. Unlike in charge-driven mixers like the SNAIL, this frequency shift is independent of the coupler Kerr, and persists even if the coupler is arrayed (dashed purple). It is possible to minimize this shift by biasing to a flux point where the coupler frequency has an inflection point as a function of DC flux (dotted black). c, Driven steady-state impurity of the LINC (left) and an equivalent SNAIL (right) as a function of drive frequency and amplitude. Both couplers are operated at a bias point where they are Kerr-free, and their parameters are optimized to match their beamsplitting strength and frequency at this bias point. Each coupler is simulated with equal decay and low-frequency dephasing. The LINC displays a significantly cleaner driven spectrum due to its parity protection.
• Figure 4: Analyzing the LINC circuita, Modes of operation of the circuit. The circuit, with an arbitrary inductive element as its center shunt, is defined by the three variables $\left\{\theta_1, \theta_2, \theta_s\right\}$. These can be re-written in terms of the more operationally relevant charge-dipole ($\hat{\theta}_c$), symmetric flux ($\hat{\phi}_{sym}$), and anti-symmetric flux ($\hat{\phi}_{asym}$) modes. b, LINC three-wave mixing strength as a function of the operating point, comparing analytic formula (grey dashed, \ref{['eq:g3wm_derivation']}) to exact time-domain and Floquet results from a squeezing operation within the LINC ($g_{\text{SL}}$, black) and a beamsplitter operation between two external resonators (teal, details in \ref{['app:FullParametric']}) respectively. The discrepancy in the beamsplitting prediction may be due to driven changes in the effective resonator-LINC participations, which are captured in the Floquet simulation but not in the analytics. c, Comparison of the analytic formulae (grey dashed, \ref{['eq:Zeeman_derivation']}) to exact Floquet simulation for driven coupler Zeeman shift as a function of DC flux, for fixed drive amplitude $\phi_{AC}=0.1\pi$.
• Figure 5: Parametric beamsplitting with the LINCa, Driven avoided crossing due to the beamsplitter interaction between two storage modes, Alice (4.9 GHz) and Bob (6.0 GHz), as a function of drive detuning from beamsplitter resonance, at a fixed drive amplitude of $\phi_{AC}=0.2\pi$. b, Driven frequency shift of each mode for an unarrayed (solid lines) vs arrayed (dashed lines) LINC. Arraying does not make a noticeable difference. c, Driven Kerr of each mode for an unarrayed (solid lines) vs arrayed (dashed lines) LINC. Arraying 3 LINCs reduces the inherited Kerr by a factor of $\sim 9$.