Two-Qubit Spectroscopy of Spatiotemporally Correlated Quantum Noise in Superconducting Qubits

Noise that exhibits significant temporal and spatial correlations across multiple qubits can be especially harmful to both fault-tolerant quantum computation and quantum-enhanced metrology. However, a complete spectral characterization of the noise environment of even a two-qubit system has not been reported thus far. We propose and experimentally demonstrate a protocol for two-qubit dephasing noise spectroscopy based on continuous-control modulation. By combining ideas from spin-locking relaxometry with a statistically motivated robust estimation approach, our protocol allows for the simultaneous reconstruction of all the single-qubit and two-qubit cross-correlation spectra, including access to their distinctive nonclassical features. Only single-qubit control manipulations and state-tomography measurements are employed, with no need for entangled-state preparation or readout of two-qubit observables. While our experimental demonstration uses two superconducting qubits coupled to a shared, colored engineered noise source, our methodology is portable to a variety of dephasing-dominated qubit architectures. By pushing quantum noise spectroscopy beyond the single-qubit setting, our work heralds the characterization of spatiotemporal correlations in both engineered and naturally occurring noise environments.

Popular Summary

Quantum information science holds the promise to deliver unprecedented computational capabilities. However, to realize this promise, the noise that affects qubit performance must be reduced. Qubits are the basic building blocks of quantum information processors, and today's qubits are limited by environmental noise for instance, stray electromagnetic fields disrupting the sensitive quantum states. While some level of noise can be corrected, it is especially difficult to deal with noise that is both time-dependent and affects more than one qubit at once. To address such correlated noise, one first must characterize it, a challenging task due to the quantum effects involved. Here, we propose and experimentally validate a protocol for achieving two-qubit quantum noise spectroscopy.

We performed a technique called spin-locking relaxometry on two superconducting qubits in the presence of correlated environmental noise. By comparing the experimentally measured behavior of the two qubits with a theoretical model, we recreated the spectral features of the noise. Our experimental validation uses superconducting qubits, but our technique is applicable to a variety of quantum architectures.

This work provides an important step towards characterizing temporal and spatial correlations in both engineered and naturally occurring noise environments.

Figure 1

Concept of the two-qubit quantum noise spectroscopy experiment. Two qubits are coupled to a common bath, leading to spatiotemporally correlated noise. The dynamics of the qubit system are modeled through a generic, broadly applicable master equation that accommodates unspecified noise self- and cross-spectra. Using spin-locking control sequences, we measure decay curves of different observables. These measured data are used to determine the noise spectra through nonlinear regression, by minimizing the deviation from the evolution of the same observables as predicted by the master equation. While our methodology is device independent, we use superconducting qubits for experimental demonstration.

Figure 2

Protocol for spectroscopy of spatiotemporally correlated noise. (a) Control and measurement cycle. The qubit system $S$ is initialized in state $\rho (0)\in \{{\rho }_s\}$, where $s\in \{1,2,\dots ,N_{\mathrm{states}}\}$ (step 1). Continuous drives (shown as orange waves) with equal Rabi frequency ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=\mathrm{\Omega }$ are then applied on the two qubits for time $t_{\mathrm{evol}}\in \{t_q\}$ with $q\in \{1,2,\dots ,N_{\mathrm{times}}\}$, during which $S$ evolves under the influence of the noise spectra evaluated at $\pm \mathrm{\Omega }$, $\{S_{jk}(\pm \mathrm{\Omega })\}$ with $j,k\in \{1,2\}$ (step 2). After this evolution time, a projective measurement of a two-qubit observable $O\in \{O_r\}$ with $r\in \{1,2,\dots ,N_{\mathrm{obs}}\}$ is performed (step 3). For a given combination $\alpha$ of initial state ${\rho }_s$, evolution time $t_q$, and observable $O_r$, this cycle is repeated $M$ times, and a sample mean ${\overline{O}}_{\alpha }$ of all outcomes $O_{\alpha }^{(m)}$ is obtained [Eq. (11)], where $m$ labels the outcome of the projective measurement for each cycle. The Rabi frequency $\mathrm{\Omega }$ is swept to gather experimental observations for all frequencies at which $S_{jk}(\pm \mathrm{\Omega })$ will be sampled. (b) Schematic of the reconstruction procedure of the two-qubit spectra from the data produced in (a). An initial value of $\mathrm{\Omega }$ is chosen, for which an initial guess of the spectrum vector $\mathbf{S}(\mathrm{\Omega })$ is assumed. This guess is fed into a nonlinear regression algorithm to find the value of $\mathbf{S}(\mathrm{\Omega })$ that globally minimizes the discrepancy between the measured sample means ${\overline{O}}_{\alpha }$ (blue circles with error bars; note, in the first plot, the occurrence of an “outlier”) and the corresponding expectation values $⟨O_{\alpha }{⟩}_{\mathbf{S}}$ obtained by numerically solving Eq. (8) for all chosen combinations $\alpha$ of the initial state, evolution time, and observable. An estimate $\hat{\mathbf{S}}({\mathrm{\Omega }}^n)$ of $\mathbf{S}({\mathrm{\Omega }}^n)$ at the $n\mathrm{th}$ Rabi frequency ${\mathrm{\Omega }}^n$ is obtained following Eq. (13) (solid red lines). The latter is then used as the initial guess for the reconstruction at the next frequency ${\mathrm{\Omega }}^{n+1}$. The procedure is repeated until $\mathbf{S}(\mathrm{\Omega })$ is reconstructed over all frequencies of interest.

Figure 3

Details of the experimental cQED device. (a) Optical micrograph of the sample. The two qubits (blue, red) are capacitively coupled to a common $\lambda /4$ resonator (green) that we use to create a shared noise source. Each qubit is also coupled to an individual resonator that we use for readout. (b) Simplified circuit diagram of our setup.

Figure 4

Ramsey interferometry experiments. (a) Ramsey sequence applied simultaneously to the two qubits, with $\pi /2$ pulses around the $y$ axis (blue boxes). A coherent drive (the “noise” tone, shown in green) is turned on well before the first pair of $\pi /2$ pulses, to bring the common resonator into a steady state with finite population, thus implementing a stationary source of engineered correlated noise. We readout both qubits simultaneously through their individual resonators, using drives at frequencies ${\omega }_{r1}$ and ${\omega }_{r2}$ (red box), thus acquiring a series of $\pm 1$ outcomes for each qubit. (b),(c) Sample mean of measured single-qubit observables $\overline{{\sigma }_{1,2}^z}$ for different wait times $t$ and different injected photon numbers $\overline{n}$ in the common cavity. The frequency of the observed Ramsey fringes varies with increasing $\overline{n}$ due to the dispersive shift $2{\chi }_j\overline{n}$ of the qubit frequencies (tilt of the vertical red and blue lines). For higher $\overline{n}$, we see a rapid dephasing of the qubit states due to the added photon shot noise (blurring of the features in the top right corners). (d) Measurement of the correlation ${\overline{C}}_{zz}=\overline{{\sigma }_1^z{\sigma }_2^z}-\overline{{\sigma }_1^z}\overline{{\sigma }_2^z}$. (e) Solution of the ME describing the qubits coupled to the common resonator [Eq. (16)], in which parameters $\kappa$, ${\chi }_j$, ${\mathrm{\Delta }}_{qj}$, and ${\gamma }_{\varphi j}$ (see Table 1) are obtained by nonlinear regression of the simulation outcomes to the experimental data. (f) Linecuts of (b) and (c) at $\overline{n}\in \{0,0.56,1.18\}$ with measured ($⧫$) and simulated (—) values of $\overline{{\sigma }_1^z}$ and $\overline{{\sigma }_2^z}$ using fitted parameters. (g) Linecuts of (d) and (e) at $\overline{n}\in \{0,0.56,1.18\}$.

Figure 5

Spin-locking experiments. (a) Spin-locking sequence applied simultaneously on two qubits with a shared source of engineered noise. (b) Bloch sphere sketch of the sequence in (a) for qubit 1 and qubit 2. The initial $\pi /2$ pulse rotates the qubits from the ground state to $|-x⟩$. The spin-locking drive effectively creates a dressed two-qubit system, with each level splitting being equal to ${\mathrm{\Omega }}_j$. Because of dephasing noise (from both the injected photon shot noise and from weaker native sources), the dressed qubits decay to their steady-state values along the $x$ axis. The final $\pi /2$ pulse turns the qubits back to the initial (${\sigma }_j^z$) quantization axis to be measured. (c) Measured correlation ${\overline{K}}_{zz}\equiv \overline{{\tau }_1^z{\tau }_2^z}-\overline{{\tau }_1^z}\overline{{\tau }_2^z}$, where the Pauli matrix ${\tau }_j^z$ is diagonal in the spin-locking basis {$|+x{⟩}_j,|-x{⟩}_j$}. The coherent drive creating photon shot noise is detuned by ${\mathrm{\Delta }}_c/2\pi =-2.03$ MHz away from the common resonator. The Rabi frequency ${\mathrm{\Omega }}_2$ of the spin-locking drive of qubit 2 is held constant at ${\mathrm{\Omega }}_2=|{\mathrm{\Delta }}_c|$, while ${\mathrm{\Omega }}_1$ is swept ($y$ axis). The correlation is significant in the region indicated by the solid black line, with width approximately $2/t$. (d) Numerical solution of the quantum-optical ME [see Eqs. (18)–(20)]. Inset: Qualitative shape of the cross-spectrum $S_{12}$ [Eq. (15)]. (e) Comparison of measured values ($⧫$) and numerical simulation ($-$) using fitted parameters. The subplots show (from left to right) ${\overline{\tau }}_1^z$, ${\overline{\tau }}_2^z$, and ${\overline{K}}_{zz}$ for ${\mathrm{\Omega }}_1/2\pi \in \{1.83,2.03\}$ MHz [indicated by black arrows in (c) and (d)]. Since ${\mathrm{\Omega }}_2$ is unchanged through the experiment, the decay curves for $\overline{{\tau }_2^z}$ are roughly the same for both linecuts. The decay curve for $\overline{{\tau }_1^z}$, however, is steeper for ${\mathrm{\Omega }}_1$ closer to the center of the photon shot-noise spectrum (top-left plot), demonstrating sensitivity to the noise frequency. The correlation ${\overline{K}}_{zz}$ roughly equals zero for ${\mathrm{\Omega }}_1\ne {\mathrm{\Omega }}_2=|{\mathrm{\Delta }}_c|$ (bottom-right plot) and shows a clear peak for ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=|{\mathrm{\Delta }}_c|$ (top-right plot). Note the negative steady-state value of the correlation for long spin-locking durations. Here, one of the dressed qubits acts as an effective decay channel for the other, with the coupling between them mediated by the noise.

Figure 6

Two-qubit quantum noise spectroscopy. (a)–(c) Representative decay curves from experimental data (circles) and the corresponding solution of the reduced ME (solid lines), Eq. (21), with estimates for the spectra $S_{jk}(\pm \mathrm{\Omega }),j,k\in \{1,2\}$, and Rabi-frequency difference $\delta \mathrm{\Omega }={\mathrm{\Omega }}_1-{\mathrm{\Omega }}_2$ obtained by nonlinear regression. The sample means are obtained with the two dressed qubits initially prepared in state $|+x,+x⟩$, and driven at the average Rabi frequency $\mathrm{\Omega }=({\mathrm{\Omega }}_1+{\mathrm{\Omega }}_2)/2=2\pi \times 2.184$ MHz (blue circles) or $1.976$ MHz (orange circles). Using the Huber loss function [Eq. (14)] makes the fitting procedure robust to outliers visible in the data [e.g., the seventh orange data point in (a)]. (d) Reconstructed two-qubit spectra. Error bars indicate 95% confidence intervals for the experimental reconstructions. Shaded orange areas indicate 95% confidence intervals associated with the ideal spectra for engineered photon shot noise obtained from Eq. (15), with parameters taken from fits described in Sec. 3: $\kappa /2\pi =198$ kHz, ${\chi }_1/2\pi =-29.1$ kHz, ${\chi }_2/2\pi =-59.5$ kHz, ${\mathrm{\Delta }}_c/2\pi =1.961$ MHz, $\overline{n}=0.127$. Blue circles indicate numerical simulation of the spectroscopy protocol using the quantum-optical ME in Eqs. (18)–(20). Inset of the bottom-right panel: Rabi-frequency difference $\delta \mathrm{\Omega }$.

Figure 7

Robust estimation approach. Comparison of spectral reconstructions using Huber loss (top row) and weighted least squares (bottom row). (a) Reconstructions of the spectra from experimental data using the Huber loss function. (b) Same reconstructions using weighted least squares. (c) Reconstructions from simulated data using the Huber loss function. (d) Same reconstructions using weighted least squares. Solid lines indicate ideal spectra given by Eq. (15). Gray, green, yellow, and brown curves correspond to $S_{11}(\omega )$, $S_{22}(\omega )$, $\mathrm{Re}[S_{12}(\omega )]$, and $\mathrm{Im}[S_{12}(\omega )]$, respectively. Inset of (b): Decay curve for the sample mean $\overline{{\tau }_2^z}$ as a function of evolution time with $\mathrm{\Omega }/2\pi =1.848$ MHz. Inset of (d): Decay curve for simulated sample means of $\overline{{\tau }_1^z}$ with $\mathrm{\Omega }/2\pi =2.120$ MHz. Blue circles represent experimental data [inset of (b)] or simulated data [inset of (d)]. Solid lines represent nonlinear regression with Huber loss (red) and weighted least squares (green). In both insets, the initial qubit state is $|+x,+x⟩$. Dashed lines in the main plots indicate spectrum frequencies $\omega /2\pi =\pm \mathrm{\Omega }/2\pi$ corresponding to the insets.