Obstacles to Continuous Quantum Error Correction via Parity Measurements

Abstract

Time-continuous quantum error correction, necessary to protect quantum information under time-dependent Hamiltonians, relies on weak continuous syndrome measurements. Implementing these measurements requires a continuous coupling among at least two qubits and a meter, a demanding requirement. We show that, under continuous operation, common parity-measurement protocols in the circuit quantum electrodynamics platform corrupt the logical information. The failure arises from approximating the three-body interaction by a sum of two-body couplings to the meter, which prevents simultaneous suppression of measurement backaction on the logical and error subspaces. We argue that the same mechanism applies more generally beyond the circuit quantum electrodynamics setting. Taken together, our results impose a practical limitation on continuous stabilizer quantum error correction and point to the viable alternatives -- architectures that realize native three-body interactions, or erasure-based encodings in which the error subspace need not be protected.

Figures (5)

• Figure 1: Continuous parity measurement scheme in the circuit QED architecture LalumierePRA10. (a) Sketch of three qubits (e.g., transmons) (top) coupled to two readout resonators (middle) to measure the stabilizers of the three-qubit bit-flip code. The two left qubits are in an even-parity state which leads to a detuning of the resonator from the probe beam and a small homodyne signal. For the odd parity of the two right qubits, the dispersive shifts cancel and the drive is on resonance which leads to a large measurement signal. (b) Contour plot of the Wigner function of the four resonator steady states, depending on the state of the qubits. (c) Time evolution of the resonator Wigner function after an error occured: Starting from an equal superposition in the odd subspace, qubit 1 undergoes a bit flip at $t=0$, which initiates evolution towards the new (even) steady state. Due to higher-order terms in the Hamiltonian, the odd steady state is shifted to negative X-values compared to (b), see \ref{['subsec:backaction_even_encoding']} for details. The parameter values for (b) and (c) are found in \ref{['tab:parameters_figures']} in \ref{['appendix:TIPT_results']}.
• Figure 2: Example runs of the continuous QEC protocol where a bit-flip error was recorded for odd (solid, orange) and even encoding (dashed, blue). The initial state is $\ket{\overline{+}}=(\ket{\overline{0}} + \ket{\overline{1}})/\sqrt{2}$ where $\ket{\overline{0}/\overline{1}}$ are the the respective logical states for the given encoding. a) Integrated photocurrent $I_\text{int}(t)$. Indicated are also the threshold values (dotted, gray) $\theta=(0.1 \overline{I}_o,0.9\overline{I}_o)$ with respect to the steady state value $\overline{I}_o=-2.8$ in the odd subspace. At the time the signal crosses the upper (lower) threshold for odd (even) encoding, here indicated by a vertical dotted orange (blue) line, an error is detected and a bit flip is applied to the faulty qubit by the controller. b) Purity $\mathcal{P}$ of the reduced state of the qubits in the respective subspace which contains the logical information, i.e., the code space in case of no error or the respective error subspace in case of an error. c) Logical phase of the encoded logical information induced by the parity measurement. For even encoding, also the estimated phase (dashed, red), calculated using \ref{['eq:phi_det', 'eq:phi_stoch']}, is shown. The parameters are found in \ref{['tab:parameters_figures']} in \ref{['appendix:TIPT_results']}.
• Figure 3: Parity measurements require three-body interactions. Left: For native three-body interactions, the states within each parity subspace are degenerate. Right: Typically native three-body interactions are not available and must be emulated by two two-body interactions. In this case, only the odd states are degenerate, whereas there is an energy splitting between the even states. This may lead to undesired entanglement with the meter as well as relative phase shifts or population differences.
• Figure 4: Is it possible to realize backaction-free continuous measurements when separating qubit-qubit coupling and qubit-meter coupling? (a) The necessary coupling between the qubits and the meter is mediated by a global coupler which, when integrated out, yields effective two-body (red) and three-body (green) interactions. (b) Additional local couplers can be used as mitigators (mit.) to address the two-body interactions which otherwise lead to undesired backaction. For the coupling scheme of Ref. MenkePRL22, the mitigator between the qubits would be optional because the two-body interaction is of $ZZ$-type, i.e, trivial for parity-encoded states.
• Figure 5: Basic idea of erasure encoding. For common encodings (left), the information or entanglement (here illustrated by wiggly lines) is preserved after an error, which makes it necessary to protect the code space and all error subspaces against measurement backaction. For erasure encoding (right), the information is erased by an error and only the code space must be protected against backaction.