Demonstrating a Long-Coherence Dual-Rail Erasure Qubit Using Tunable Transmons

Quantum error correction with erasure qubits promises significant advantages over standard error correction due to favorable thresholds for erasure errors. To realize this advantage in practice requires a qubit for which nearly all errors are such erasure errors, and the ability to check for erasure errors without dephasing the qubit. We demonstrate that a “dual-rail qubit” consisting of a pair of resonantly coupled transmons can form a highly coherent erasure qubit, where transmon $T_1$ errors are converted into erasure errors and residual dephasing is strongly suppressed, leading to millisecond-scale coherence within the qubit subspace. We show that single-qubit gates are limited primarily by erasure errors, with erasure probability $p_{\text{erasure}}=2.19(2)\times {10}^{-3}$ per gate while the residual errors are $\sim 40$ times lower. We further demonstrate midcircuit detection of erasure errors while introducing $<0.1\%$ dephasing error per check. Finally, we show that the suppression of transmon noise allows this dual-rail qubit to preserve high coherence over a broad tunable operating range, offering an improved capacity to avoid frequency collisions. This work establishes transmon-based dual-rail qubits as an attractive building block for hardware-efficient quantum error correction.

Popular Summary

Achieving the promise of quantum computing will likely require error correction, whereby many noisy physical qubits are used to encode a smaller number of robust logical qubits. This overhead, in qubit number and complexity, is a major obstacle to scaling up quantum computers. In this work, we experimentally demonstrate a recently proposed type of qubit, coined an “erasure qubit.” In contrast to standard qubits, erasure qubits are designed to directly flag errors as they occur. This extra information about the presence of errors in the system could allow for substantially improved error-correction performance and lower overhead.

We engineer erasure qubits out of standard building blocks called transmons. In this context, transmons can be understood as elements that can hold single photons, but with the challenge that the photons eventually leak out. To ensure that such photon loss can be flagged, we use a pair of transmons and encode a “dual-rail” qubit by having a single photon in either the left transmon or the right transmon. By checking to see if a photon remains in the system, we can detect whether photon loss occurred. Furthermore, by hybridizing the transmons, we make this system robust to other forms of noise that normally affect transmons. We experimentally show that this system is an effective erasure qubit, with minimal errors beyond the flagged photon loss.

Our work suggests an exciting path for both completing the error-correction toolbox with erasure qubits, and for scaling up to larger systems. This promising research direction can help to accelerate the future of large-scale, useful quantum computation.

Figure 1

Dual-rail erasure qubit encoding. (a) Our circuit has three tunable transmons: Q1 and Q2 form the dual-rail qubit, and Q3 is an ancilla qubit used for erasure detection (insets: tuning ranges, with primary operating point marked). (b) Microwave spectroscopy of Q1 as Q2 is tuned into resonance, exhibiting an avoided crossing with gap $2g/2\pi =180\mathrm{MHz}$. (c) The dual-rail subspace is the symmetric and antisymmetric combination of $|01⟩$ and $|10⟩$. Decay of the underlying transmons leads to detectable leakage to $|00⟩$, constituting an erasure error. (d) Circuits begin with a 40 ns microwave pulse to initialize $|1_L⟩$, followed by single-qubit gates enacted by flux modulation of Q2, and finally the qubits are adiabatically separated and jointly read out. (e) We perform spectroscopy on the ancilla after initializing the dual-rail pair in $|00⟩$ (blue), $|1_L⟩$ (orange), or $|0_L⟩$ (green, trace not shown). Erasure checks are performed by applying a microwave pulse on the ancilla which is resonant only if the dual-rail pair is in $|00⟩$ (blue vertical line). (f) Spin-echo experiment on the dual-rail qubit where the phase of the final $\pi /2$ is stepped, showing both coherent fringes within the subspace and also decay out of the subspace with a lifetime of $\sim 30\mathrm{\mu }\mathrm{s}$. Error bars denote 68% confidence intervals unless otherwise specified.

Figure 2

Millisecond-scale coherence within the dual-rail subspace. (a) We measure $T_2$ coherence with a CPMG (Carr-Purcell-Meiboom-Gill) sequence [32] consisting of $N$ $\pi$ pulses with a total evolution time of $N\tau$. In parallel, we perform a sequence of $N_e=11$ erasure checks, evenly spaced by ${\tau }_e=N\tau /(N_e-1)$. Coherence data are postselected against leakage using both the final dual-rail readout and also the midcircuit erasure checks. (b) Example experiment with $N=64$ $\pi$ pulses, where the final $\pi /2$ pulse phase is stepped to produce fringes. The postselection probability decays with time according to a $29.3(4)\mathrm{\mu }\mathrm{s}$ erasure lifetime, while the postselected shots are fit to an exponential decay with extrapolated $T_2=849(51)\mathrm{\mu }\mathrm{s}$. 330 000 shots were taken per point, but only a small fraction of these survive postselection. (c) CPMG measurements on each underlying transmon, if parked alone at the operating point, as well as on the dual-rail qubit. Error bars are standard deviation of many individual measurements. (d) $T_1$ within the dual-rail subspace. We measure the probability of ending in $|1_L⟩$, postselected against erasure errors, after initializing in either $|1_L⟩$ (blue) or $|0_L⟩$ (orange). The difference between these traces is fit to an exponential decay with fixed offset 0, giving $T_1=906(15)\mathrm{\mu }\mathrm{s}$. The 24 hour time trace of data comprising these dual-rail $T_1$, $T_2$ plots is shown in Appendix pp12.

Figure 3

Single-qubit randomized benchmarking. (a) The randomized benchmarking circuit uses random Cliffords, each implemented as two X90 pulses with appropriate phase shifts [33]. In parallel, we perform 11 erasure checks evenly spaced throughout the circuit, regardless of depth. The X90 pulse is a Gaussian-shaped 48 ns flux pulse on Q2 along with a 0.067 rad phase correction. An example timing diagram is presented in Fig. 9. (b) We plot the postselection probability (green) as a function of circuit depth and fit to an exponential decay to 0 to extract the erasure error per Clifford of $4.38(3)\times {10}^{-3}$. The postselected survival probability within the dual-rail subspace (blue), averaged over 200 random circuits for each depth, is fit to an exponential decay with offset $1/2$ and gives a residual error rate of $1.01(1)\times {10}^{-4}$ per Clifford. Individual outcomes for random circuits are plotted with transparency. We symmetrize readout errors by alternating measurements in which the ideal outcome is $|0_L⟩$ or $|1_L⟩$. The X90 gate errors are half those of the Clifford gate, with erasure error $2.19(2)\times {10}^{-3}$ and residual error $5.06(6)\times {10}^{-5}$. Similar error rates are computed when postselecting only on midcircuit erasure checks (Appendix pp6).

Figure 4

Characterizing midcircuit erasure detection. (a),(b) We characterize false-positive and false-negative errors by initializing a target state, either $|1_L⟩$ or $|00⟩$, performing an erasure check, and then analyzing its results postselected on a final readout confirming the correct initialization. We ensure initialization of the ancilla in $|0⟩$ by postselecting on an additional prereadout before the sequence begins (not shown). (c) We characterize erasure-check-induced dephasing by inserting a variable number of checks into a spin-echo experiment with fixed total evolution time $T=114\mathrm{\mu }\mathrm{s}$. The $N$ erasure checks are evenly spaced with separation time ${\tau }_e=T/N$. We vary the phase of the final $\pi /2$ pulse to measure the remaining degree of coherence, postselected on all erasure checks indicating no erasure. Coherence appears to grow with a small number of checks due to properly catching and postselecting against leakage into $|00⟩$ which reheats into the subspace. This effect saturates, after which we observe little dephasing per check at the level of $<0.1\%$ (see Appendix pp9). The postselection probability (right-hand subplot) decreases with increasing erasure checks, first due to discarding shots with leakage, then saturating at a rate of $\sim 0.8\%$ per check which we interpret as a combination of false-positive assignment errors and check-induced erasure events. We note that a positive erasure detection event can with low probability, 0.0293(3)%, reexcite the dual-rail qubit (see Appendix pp2).

Figure 5

Robust operation away from sweet spot. (a) The dual-rail qubit can be operated anywhere the two transmons can be brought onto resonance, forming an engineered “tunable” sweet spot. (b) At a range of operating points, we measure $T_2^{\mathrm{echo}}$ of each underlying transmon individually as well as of the dual-rail qubit. With the exception of a narrow region around 4.96 GHz, the dual-rail qubit maintains a coherence of several hundred microseconds over most of this range [$366(20)–550(30)\mathrm{\mu }\mathrm{s}$], with the modest degradation likely due to increased effects of Q1’s flux noise away from its sweet spot (Appendix pp14). These coherence times are over an order of magnitude larger than the higher-coherence transmon (Q1) and more than 2 orders of magnitude larger than Q2. The dip in dual-rail $T_2$ at 4.96 GHz is likely explained by a TLS coupled to Q2 which is near resonant with the upper hybrid mode at this operating point; a similar reduction is present on Q2 directly at an offset operating point, as visible in supplementary datasets presented in Appendix pp15. Error bars are standard deviations of many fit results (individual dots) for Q1 and Q2, and are fit uncertainties for the dual-rail qubit.

Figure 6

Optical image of superconducting device. False-color overlays indicate the functional purpose of each component needed to define the transmon qubits, control them with microwave and flux lines, and perform readout.

Figure 7

Controls and cryogenic wiring. We show the signal chain including components at various temperature stages within the dilution refrigerator. A single readout chain is used for the processor. Each qubit has an $XY$ control line, as well as a slow-flux line which is combined with a fast-flux line on a bias tee with the capacitor removed from the fast-flux side to enable low-frequency flux pulses. The fast-flux line for Q2 is generated by combining a signal which is low-pass filtered (cutoff $<22\mathrm{MHz}$), used for flux-assisted readout, along with another source which is attenuated and high-pass filtered (cutoff $>41\mathrm{MHz}$), used for flux modulation at $2g=2\pi \times 180\mathrm{MHz}$ to drive single-qubit gates; this setup mitigates high-frequency control noise which could limit the dual-rail $T_1$. The fast-flux lines for the other qubits (Q1 and Q3) are not needed for these experiments and are terminated with $50\mathrm{\Omega }$ at room temperature (not shown).

Figure 8

Dual-rail calibration procedure. (a) After parking the transmons on resonance, we perform microwave spectroscopy to identify the two hybrid modes $|0_L⟩$ and $|1_L⟩$ at ${\omega }_0\pm g$. A small additional feature at ${\omega }_0-\eta /2$ is observed, which is a two-photon resonance going to higher excited states; its near overlap with the transition to $|0_L⟩$ is particular to these parameters where $g\sim \eta /2$, so for this device we choose to initialize $|1_L⟩$. Inset: excitation probability as we scan the pulse amplitude on the $|1_L⟩$ resonance. (b) We calibrate flux-modulation gates by applying a flux-modulation pulse of variable amplitude in between two microwave pulses; this measurement detects population transfer from $|1_L⟩$ to $|0_L⟩$ induced by the flux pulse, and is used to approximately calibrate a $\pi /2$ or $\pi$ pulse within the dual-rail subspace. (c) A Ramsey experiment used to measure the dual-rail frequency relative to a reference frequency. We show here the full bit-string probabilities of the Q1 and Q2 readout; the 01 and 10 results are interpreted as representing population in $|0_L⟩$ and $|1_L⟩$, respectively, prior to the adiabatic separation of qubits during readout.

Figure 9

Experimental sequence for randomized benchmarking. The sequence begins with $\sim 350\mathrm{\mu }\mathrm{s}$ idle time for the transmons to relax into $|0⟩$. A microwave pulse on Q1 initializes the dual-rail qubit in $|1_L⟩$. A sequence of Cliffords is then applied, each composed of two X90 pulses with a particular phase shift of the drive field. Shown here is a sequence with 400 Cliffords (800 X90s). In parallel with this sequence, 11 erasure checks are performed, evenly spaced across the full circuit. Finally, a baseband flux pulse is applied to Q2 to adiabatically separate Q1 and Q2, during which these qubits are read out. FF stands for fast flux.

Figure 10

Randomized benchmarking postselection protocols. (a) The final dual-rail readout, postselected only on midcircuit erasure checks (left-hand panel), shows a gradual decline in the ideal outcome $P(10)$, as well as a background of $P(00)\approx 10\%$ due to readout errors. The $P(10)$ trace does not decay to a known offset, but a fit to $A{e}^{-pN}$ is valid for short times and gives an error of $p=1.10(2)\times {10}^{-4}$ per Clifford. This result is close to the error rate evaluated with additional postselection against $|00⟩$ and $|11⟩$ in the final readout (right-hand panel), which as reported in the main text has error $1.01(1)\times {10}^{-4}$. (b) The erasure error per Clifford is computed based on the decay of postselection probability with increasing circuit depth, and gives nearly the same fitted erasure probability per Clifford when using just midcircuit erasure checks [0.437(3)%] as when using both midcircuit checks and final readout as reported in the main text [0.438(3)%].

Figure 11

Deterministic phase shift induced by erasure check. (a) In a similar experiment to that presented in Fig. 4, we perform a spin echo and insert a variable number of erasure checks into either the first half, the second half, or both halves of the sequence. (b) We see that the phase of the dual-rail qubit is essentially independent of the number of erasure checks when balanced across both arms, but picks up a phase shift proportional to the number of checks when all checks are applied within one arm. The phase accumulation per check fits to 0.0142(3) rad. (c) We illustrate example phase measurements of the points circled in (b), in which we scan the phase of the final $\pi /2$ pulse in the spin-echo sequence and fit to a sinusoidal model.

Figure 12

Varying the number of erasure checks during coherence measurements. When measuring dual-rail $T_2$ or $T_1$ with no erasure checks, the coherence appears to rapidly decay over $200–300\mathrm{\mu }\mathrm{s}$ due to leakage to $|00⟩$ which then heats back into the dual-rail subspace. As we add more erasure checks (evenly spaced throughout the sequence for each variable length), we find that 11 erasure checks is enough to eliminate the contribution of such instances so that we can more accurately probe coherence within the subspace.

Figure 13

Telegraph noise on the dual-rail qubit. (a) We show two typical Ramsey $T_2^{*}$ measurements, integrated for $\sim 1\min$. Within the envelope of not decaying to $|00⟩$, we see fringes which track the phase coherence of the dual-rail qubit. We find in some cases stable fringes (upper plot) and in other cases decay-revival behavior (lower plot). (b) By repeatedly measuring the dual-rail frequency over shorter time intervals, we identify telegraphlike switching in the frequency. Integrating a Ramsey experiment for longer than this switching time gives rise to the beating effect seen in (a). By interleaving this time-resolved measurement with similar time-resolved measurements of Q1’s frequency and flux-pulse spectroscopy measurements on Q1, we note that this switching is present in similar scales on both the dual-rail and Q1 and is tied to the toggling of a TLS-like mode at 4.98 GHz. When the TLS is present, it reduces the readout fidelity of Q1 leading to a reduced contrast in the spectroscopy slice.

Figure 14

Stability of dual-rail coherence metrics. The coherence measurements presented in Fig. 2 were averaged over data acquired over $\sim 24$ h. Here, we analyze subsets of the same dataset in a time-resolved way to track the fitted coherence metrics during that period. We show three representative metrics: (a) CPMG with one $\pi$ pulse, (b) CPMG with $64\pi$ pulses, and (c) $T_1$ within the dual-rail subspace. In each plot, we additionally show the erasure lifetime extracted during each measurement. For the $T_1$ measurement, this erasure lifetime is extracted separately for the two logical states and found to be fluctuating differently for the two states, despite the measurements being taken in an interleaved fashion. Large discrepancies between the erasure lifetimes of the two logical states can lead to dephasing of the dual-rail qubit, but this effect is mitigated by dynamical decoupling.

Figure 15

Measuring dependence of dual-rail frequency on transmon flux. At the two extremal operating points from Fig. 5, we measure how the dual-rail frequency shifts as we shift the flux on one of the two transmons away from its calibrated operating point (upper row). Each point here is measured with a Ramsey experiment on the dual-rail qubit. The functional form is dependent both on how the dual-rail frequency changes with underlying transmon frequency and also how the transmon frequency shifts with flux. By independently calibrating how the transmon frequencies change in this range, we plot the same data with an altered $x$ axis that reflects instead how much the transmon frequency shifted (bottom row). In both rows, the transparent trace is the analytic model from Eq. (m3), with the only adjusted parameters being the calibrated dual-rail energy gap at each operating point.

Figure 16

Supplementary coherence data across tunable operating band. We show $T_2^{*}$, $T_2^{\mathrm{echo}}$, and $T_1$ for both transmons Q1 and Q2 at their range of operating points. For the dual-rail qubit, we show the $T_2^{\mathrm{echo}}$ as well as the erasure lifetime for each logical state. We find that the dip in dual-rail coherence at 4.96 GHz corresponds to a drop in the $|1_L⟩$ erasure lifetime; this upper hybrid mode is at $4.96\mathrm{GHz}+90\mathrm{MHz}=5.05\mathrm{GHz}$. We attribute this feature to a TLS coupled to Q2, which is independently seen as the sharp drop in Q2’s $T_1$ around 5.05 GHz. All individual points are fit results for single measurements; for the transmon data for Q1 and Q2, the average result (error bar) is the median of the individual datasets (standard deviation). For the dual-rail $T_2$ data, the average (error bar) is the fit result of all datasets averaged together (fit uncertainty).