Low-Latency Digital Signal Processing for Feedback and Feedforward in Quantum Computing and Communication

Quantum computing architectures rely on classical electronics for control and readout. Employing classical electronics in a feedback loop with the quantum system allows us to stabilize states, correct errors, and realize specific feedforward-based quantum computing and communication schemes such as deterministic quantum teleportation. These feedback and feedforward operations are required to be fast compared to the coherence time of the quantum system to minimize the probability of errors. We present a field-programmable-gate-array-based digital signal processing system capable of real-time quadrature demodulation, a determination of the qubit state, and a generation of state-dependent feedback trigger signals. The feedback trigger is generated with a latency of 110 ns with respect to the timing of the analog input signal. We characterize the performance of the system for an active qubit initialization protocol based on the dispersive readout of a superconducting qubit and discuss potential applications in feedback and feedforward algorithms.

Figure 1

(a) Overview of the feedback loop. Typical latencies are indicated in blue, and typical carrier frequencies of the signal are indicated in gray. See the text for details. (b) Sketch of the time dependence of the in-phase component $I$ of the readout signal, which approaches different steady-state values depending on whether the qubit is in state $|g⟩$ (the blue curve) or $|e⟩$ (the red curve). We consider a scenario in which the response time of the resonator is much shorter than the lifetime of the qubit. Specific times indicated are the onset of the readout pulse ($t_0$), as well as the beginning ($t_1$) and the end ($t_2$) of the integration time (${\tau }_i$, the blue shaded region). We define the total readout time ${\tau }_{\mathrm{RO}}$ as the time difference between $t_0$ and $t_2$ (the blue arrow between the dashed lines) [44]. (c) Sketch of the trajectories in the plane spanned by the $I$ and $Q$ components of the signal for the states $|g⟩$ (the blue curve) and $|e⟩$ (the red curve). Specific points in the trajectories are marked corresponding to the times $t_0$, $t_1$, and $t_2$ as defined in (b). (d) Sketch of the typical distribution of the integrated in-phase component ($I$) when the qubit is in state $|g⟩$ (the blue curve) or $|e⟩$ (the red curve). The dashed line represents the threshold value $I_t$ for determining the state of the qubit.

Figure 2

Overview of the digital signal processing circuit showing the flow of the digitized signal (the black arrows) and the trigger lines (the orange arrows). The symbols $z^{-n}$ denote delays by $n$ clock cycles implemented with synchronous $D$ flip-flops. The blue dashed lines mark positions at which the signal is further registered in pipelined registers not explicitly shown. The corresponding latencies of the pipeline stages are written below the blue arrows. The dotted lines indicate settings defined via the interface with the host computer. Explanations of each circuit block are given in the text.

Figure 3

Calculated signals at different processing stages for exemplary inputs when the qubit is in either the ground state (the blue line) or the excited state (the red line). The blue squares and the red diamonds represent the corresponding simulated digital signals obtained from a simulation of the FPGA design. The vertical axes display arbitrary units. The blue arrows and the dashed lines visualize the delays of the signals relative to each other. (a) The signals $V_{\mathrm{ADC}}$ from the ADC with two different phases, depending on whether the qubit is in the ground (blue) or excited state (red), together with the corresponding trigger (tr.) signals (the orange line). (b) Real ($\mathrm{Re}[S_m]$) and imaginary ($\mathrm{Im}[S_m]$) part of the complex signal at the output of the digital mixer with the corresponding trigger delayed by one clock cycle ($z^{-1}\mathrm{tr}.$). (c) In-phase ($I$) and quadrature ($Q$) component of the signal obtained at the output of the FIR filter corresponding to a moving average of four consecutive points, with the corresponding trigger delayed by two clock cycles ($z^{-2}\mathrm{tr}.$). (d) Feedback trigger (FBT) conditioned on a threshold on $I$ indicated by the thick horizontal bar at $t=160\mathrm{ns}$, which is set by the user-definable delay $z^{-d}$ of $d=14$ clock cycles.

Figure 4

(a) Quantum circuit depicting the experimental protocol to test the feedback routine. The state at each stage of the protocol is represented on the Bloch sphere (blue). The horizontal black line indicates the evolution of the qubit state over time. Double arrows ($\Rightarrow$) represent the flow of classical information. The sequence of operations is a $\pi /2$ rotation (the green box) of the Bloch vector about an equatorial axis, a first projective measurement ($M1$), a conditional $\pi$ rotation (the red dashed box) that depends on the feedback trigger (FBT) determined by the digital signal processing (DSP), and a second projective measurement ($M2$) of the qubit state. (b) Pulse scheme showing the timing of microwave pulses applied to the qubit (the green trace), the pulses applied to the resonator (the yellow trace), and the conditional $\pi$ pulse applied to the qubit (the red dashed line). The blue shaded regions mark the integration windows of the measurements $M1$ and $M2$. The time offset ${\tau }_{\mathrm{RO}}$ marks the time from the beginning of each readout pulse to the end of the corresponding integration window, ${\tau }_{\mathrm{EL},\mathrm{tot}}$ (the blue arrow) marks the delay in the feedback electronics, and ${\tau }_{\mathrm{FB}}$ marks the total feedback latency as it is defined in the text. (c) Histograms of the in-phase signal $I_1$ obtained from the first readout pulse $M1$ (the blue dots) and in-phase signal $I_2$ obtained from the second readout pulse $M2$ (the orange dots) for the case when the feedback actuator is disabled. The dashed line marks the feedback threshold. For $M1$ and $M2$, the percentage of counts on the right side of the threshold is indicated. (d) The same type of histograms as in (c) but with the feedback actuator enabled. (e) Two-dimensional histogram with $128\times 128$ bins counting the combined outcomes of the first readout $I_1$ (the horizontal axis) and the second readout $I_2$ (the vertical axis) for the case when the feedback actuator is disabled. The plane is divided into four regions (${\mathcal{R}}_{GG}$, ${\mathcal{R}}_{GE}$, ${\mathcal{R}}_{EG}$, and ${\mathcal{R}}_{EE}$) separated by the threshold (the dashed lines). The percentage of counts relative to the total count is indicated in each quadrant. Red lines are contour lines marking specific counts of $\{0.5,1,2\}\times {10}^3$. (f) The same type of two-dimensional histogram as in (e) but with the feedback actuator enabled.

Figure 5

Schematic of the experimental setup used for quantum feedback. Cyan arrows point into the direction of the signal flow in the feedback loop. The device under test (DUT) is a superconducting circuit comprising a qubit coupled to a coplanar waveguide resonator (CPWR). The color scheme of the blocks, DUT (green), static control (gray), detector (yellow), Nallatech BenADDA-V4 card (blue), and actuator (red), corresponds with Fig. 1. In addition, the three different shades of purple in the dilution refrigerator (the purple box) indicate the temperature stages (4-K plate, 100-mK plate, and baseplate) to which the corresponding components are thermalized. The signal ports at the dilution refrigerator are labeled with letters in circles ($A–E$). The AWG 1 in the static control unit has four channels (CH 1–4) while the AWG 2 in the actuator has two channels (CH 1–2).

Figure 6

Details of the blocks of the DSP circuit relevant to feedback generation. (a) Digital $I\text{−}Q$ mixer implemented with multipliers (the circles with crosses). (b) Quarter sampling rate $f_s/4$ digital $I\text{−}Q$ mixer implemented with multiplexers (MUXs) that forward one of their inputs to their output based on the selection (sel.). The sel. signal is driven by a repeating two-bit counter (CNT,$++$). (c) Moving average using an accumulator ($+=$). (d) Circuit for offset subtraction and scaling. (e) Schematic of the logic circuit of the state discrimination module based on lookup tables (LUTs). (f) Representation of the plane spanned by the offset-subtracted in-phase ($\tilde{I}$) and quadrature ($\tilde{Q}$) signal components. The four quadrants are labeled with the corresponding values of the sign bits $x$ and $y$. (g) Representation of one LUT based on the inputs $x$ and $y$ (the first two columns). The third column contains the symbolic value $L_{xy}$ stored in the LUT for every combination of input bits $x$ and $y$. (h) Specific example of how to fill in the LUT.

Figure 7

(a) Sample circuit for recording two-dimensional histograms whose dimensions are the preprocessed 7-bit signals $\tilde{I}$ and $\tilde{Q}$. See the text for details. (b) Correlation mode with buffer and segment counter (seg. cnt). For simplicity, the rounding steps are not shown here. (c) Time-resolved mode with a time counter (time cnt). For simplicity, the rounding steps are not shown here. (d) Illustration of exemplary time-resolved histograms of the $\tilde{I}$ and $\tilde{Q}$ values at four consecutive times $t$ when the qubit state is $|g⟩$ (blue) or $|e⟩$ (red).

Figure 8

Same type of histograms as presented in Fig. 4 for the scenario where the initial $\pi /2$ pulse is omitted. (a) Histograms of the in-phase signal in the first measurement $I_1$ (the blue dots) and second measurement $I_2$ (the orange dots) when feedback is disabled. The dashed line marks the feedback threshold. Percentages are the summed counts of occurrences above the threshold relative to the total count $C_{\mathrm{tot}}=2097152$ for the signals in $M1$ and $M2$, respectively. (b) The same type of histograms as in (a) but with feedback enabled. (c) Two-dimensional histogram with $128\times 128$ bins as a function of the in-phase signal in the first measurement $I_1$ versus the in-phase signal in the second measurement $I_2$ with feedback disabled. The red lines are contour lines marking specific counts of $\{0.05,1,2,4\}\times {10}^3$. In each region (${\mathcal{R}}_{GG}$, ${\mathcal{R}}_{GE}$, ${\mathcal{R}}_{EG}$, and ${\mathcal{R}}_{EE}$) separated by the threshold (the dashed lines), the percentage of counts relative to the total count is indicated. (d) The same type of two-dimensional histogram as in (c) but with feedback enabled.