Periodic Reporting for period 1 - DANCINGFOOL (High-impedance Superconducting Circuits Enabling Fault-tolerant Quantum Computing by Wideband Microwave Control)
Reporting period: 2022-12-01 to 2025-05-31
A physical system implementing a quantum bit (qubit) is never perfectly isolated from an uncontrolled environment. The system dynamics is thus noisy, modifying randomly the qubit state. This phenomenon of decoherence is the main roadblock to build a stable quantum computing platform. In order to mitigate decoherence, quantum error correction employs only a few code states within a much larger informational space, so that noise-induced dynamics can be detected and corrected before the encoded information gets corrupted. Unfortunately, most known protocols require to control dauntingly complex systems.
Our project is to build an autonomously error-corrected qubit encoded in the vast phase-space of a quantum oscillator, implemented by a resonant mode of a superconducting circuit. In particular, we focus on qubits encoded in the form of Gottesman-Kitaev-Preskill (GKP) states, which have the unique ability to correct both types of quantum errors (bit and phase flips) induced by the dominant noise channels. In prior attempts to stabilize and correct GKP qubits, one couples the target oscillator embedding the GKP qubit to an auxiliary physical qubit in order to detect errors. This strategy allows noise to propagate from the auxiliary qubit to the target oscillator, opening uncorrectable error channels.
We follow a radically different approach to stabilize and correct GKP states. Our strategy is more demanding as it requires to simultaneously activate multiple parametric processes between a high-impedance circuit mode and a dissipative environment---a strategy known as dissipation engineering. An important technical challenge is that the target mode should remain purely harmonic beside these controls. On the flip side, once this regime has been reached, the target mode autonomously stabilizes in the GKP code manifold and noise from the dissipative environment cannot propagate to the encoded qubit.
Our project is to build an autonomously error-corrected qubit encoded in the vast phase-space of a quantum oscillator, implemented by a resonant mode of a superconducting circuit. In particular, we focus on qubits encoded in the form of Gottesman-Kitaev-Preskill (GKP) states, which have the unique ability to correct both types of quantum errors (bit and phase flips) induced by the dominant noise channels. In prior attempts to stabilize and correct GKP qubits, one couples the target oscillator embedding the GKP qubit to an auxiliary physical qubit in order to detect errors. This strategy allows noise to propagate from the auxiliary qubit to the target oscillator, opening uncorrectable error channels.
We follow a radically different approach to stabilize and correct GKP states. Our strategy is more demanding as it requires to simultaneously activate multiple parametric processes between a high-impedance circuit mode and a dissipative environment---a strategy known as dissipation engineering. An important technical challenge is that the target mode should remain purely harmonic beside these controls. On the flip side, once this regime has been reached, the target mode autonomously stabilizes in the GKP code manifold and noise from the dissipative environment cannot propagate to the encoded qubit.
Our project is organized along three complementary axes explored in parallel.
In the first axis, we design and test the hardware to stabilize GKP qubits by dissipation engineering. In term, the challenge is to control with multiple high-order parametric interactions a superconducting resonator of high-impedance and low-frequency. One of the challenges is that the resonator should otherwise remain perfectly linear.
We have tested several prototype circuits toward this goal. As a proof of concept, we first targeted a moderate impedance resonator (Z~1 kOhm) which is an order of magnitude lower than required to stabilize GKP qubits, but still an order of magnitude higher than typical circuits employed in dissipation engineering. With this circuit, we parametrically activated a high-order multiphotonic interaction between the target mode and a dissipative environment, by which quartets of excitations from the former are dissipated in the latter. The main challenge that we faced is that several mechanisms, which can be safely neglected in low impedance circuits, yield a significant non-linearity of the target mode in the regime we explored. As a consequence, the target mode is no longer harmonic and the multiphotonic interaction is only resonant for a specific state of the resonator. In the near future, we plan on adapting the circuit design to damp spurious non-linearities, allowing for the stabilization of a four-legged Scrödinger cat state. This would be a major milestone for quantum error correction, and would pave the way for the stabilization of GKP qubits.
In the second axis, we laid out solid mathematical foundations for our project. After deriving the target dynamics that efficiently stabilize GKP qubits, we showed how to obtain these dynamics in a high-impedance circuit parametrically driven with multiple high-order parametric processes. We analyzed the impact of various imperfections in a realistic system, and showed that errors could be suppressed far beyond the state-of-the-art with our methods.
In the third axis, we developed on-chip microwave components such as microwave switches and circulators. The latter are devices in which microwaves are routed to different ports depending on their direction of propagation. Designing reliable microwave components that can be incorporated to the quantum system of interest (“on-chip”) is crucial for the scalability of superconducting processors. For our project which requires to address a system with parametric drives covering a wide-frequency band, tailoring these components to our needs could greatly facilitate the system operation.
In practice, we designed a novel two-mode traveling wave Josephson device in which a pump wave enables the frequency conversion of a probe wave. Compared to conventional traveling-wave device, the pump travels on a distinct mode and at lower speed than the probe. This allows us to convert the probe signal into an idler wave propagating in the reverse direction, providing a more robust conversion process. Adjusting the pump frequency and direction of propagation, one configures the device to activate the desired functionality (microwave switch or on-chip circulation) on the desired band.
In the first axis, we design and test the hardware to stabilize GKP qubits by dissipation engineering. In term, the challenge is to control with multiple high-order parametric interactions a superconducting resonator of high-impedance and low-frequency. One of the challenges is that the resonator should otherwise remain perfectly linear.
We have tested several prototype circuits toward this goal. As a proof of concept, we first targeted a moderate impedance resonator (Z~1 kOhm) which is an order of magnitude lower than required to stabilize GKP qubits, but still an order of magnitude higher than typical circuits employed in dissipation engineering. With this circuit, we parametrically activated a high-order multiphotonic interaction between the target mode and a dissipative environment, by which quartets of excitations from the former are dissipated in the latter. The main challenge that we faced is that several mechanisms, which can be safely neglected in low impedance circuits, yield a significant non-linearity of the target mode in the regime we explored. As a consequence, the target mode is no longer harmonic and the multiphotonic interaction is only resonant for a specific state of the resonator. In the near future, we plan on adapting the circuit design to damp spurious non-linearities, allowing for the stabilization of a four-legged Scrödinger cat state. This would be a major milestone for quantum error correction, and would pave the way for the stabilization of GKP qubits.
In the second axis, we laid out solid mathematical foundations for our project. After deriving the target dynamics that efficiently stabilize GKP qubits, we showed how to obtain these dynamics in a high-impedance circuit parametrically driven with multiple high-order parametric processes. We analyzed the impact of various imperfections in a realistic system, and showed that errors could be suppressed far beyond the state-of-the-art with our methods.
In the third axis, we developed on-chip microwave components such as microwave switches and circulators. The latter are devices in which microwaves are routed to different ports depending on their direction of propagation. Designing reliable microwave components that can be incorporated to the quantum system of interest (“on-chip”) is crucial for the scalability of superconducting processors. For our project which requires to address a system with parametric drives covering a wide-frequency band, tailoring these components to our needs could greatly facilitate the system operation.
In practice, we designed a novel two-mode traveling wave Josephson device in which a pump wave enables the frequency conversion of a probe wave. Compared to conventional traveling-wave device, the pump travels on a distinct mode and at lower speed than the probe. This allows us to convert the probe signal into an idler wave propagating in the reverse direction, providing a more robust conversion process. Adjusting the pump frequency and direction of propagation, one configures the device to activate the desired functionality (microwave switch or on-chip circulation) on the desired band.
The results obtained along the three axes described above are beyond the state-of-the-art:
- A high-order multiphotonic dissipation channel by which a system dissipates quartets of excitations (experiment). The possibility to activate such high-order channels opens new avenues for the stabilization of advanced bosonic qubits
- A novel approach to stabilize GKP qubits avoiding noise propagation during error-syndrome extraction (theory). Noise propagation is the main limit to current GKP qubits.
- A two-mode travelling wave device allowing the mixing of counterpropagating waves in a controlled manner (theory and experiment). This configuration yields more robust wave conversion than the conventional one in which all waves propagate in the same direction.
- A high-order multiphotonic dissipation channel by which a system dissipates quartets of excitations (experiment). The possibility to activate such high-order channels opens new avenues for the stabilization of advanced bosonic qubits
- A novel approach to stabilize GKP qubits avoiding noise propagation during error-syndrome extraction (theory). Noise propagation is the main limit to current GKP qubits.
- A two-mode travelling wave device allowing the mixing of counterpropagating waves in a controlled manner (theory and experiment). This configuration yields more robust wave conversion than the conventional one in which all waves propagate in the same direction.