Constant depth magic state cultivation with Clifford measurements by gauging

Abstract

Magic states are a scarce resource for two-dimensional qubit stabilizer codes. Magic state cultivation was recently proposed to reduce the cost of magic state preparation by measuring the transversal Clifford operator of the color code. Cultivation achieves $\sim 10^{-9}$ logical error rates for the $d=5$ color code, with substantially lower space-time overhead than magic state distillation. However, due to the $\mathcal{O}(d)$ depth of the Clifford measurement circuit, magic state cultivation becomes impractical for $d>5$. Here, we perform logical $XS^\dagger$ measurements on the color code by gauging a transversal Clifford gate, resulting in a constant-depth logical measurement circuit. We employ repeated gauging measurements with post-selection rather than performing error correction on the Clifford stabilizer code that emerges during the gauging protocol, thus gaining simplicity at the cost of scalability. Our protocol requires a regular square grid connectivity and yields logical error rates comparable to magic state cultivation. The $d=7$ version of our protocol gives access to the $10^{-12}$ logical error rate regime at $0.05\%$ physical error rate while retaining more than $1\%$ of the shots after the equivalent of the cultivation stage.

Figures (6)

• Figure 1: (a) Gauging measurement of the logical $XS^\dagger$ operator supported on data qubits ($d$ and $d'$) using ancillas ($a$ and $a'$) on the edges of the honeycomb lattice. The weight of logical Z operator (cyan) is reduced during the measurement. (b) Gauge operators for measuring the transversal $XS^\dagger$ on the $d=3$ color code. (c) Two minimum-weight logical representatives (solid and dashed cyan lines) of the $d=7$ color code. Arrows point at the flag qubit shared by the same colored edges. (d) Flagged version of the circuit in (a), where an extra qubit ($f$) increases the weight of the logical before the the circuit distance would drop.
• Figure 2: (a) Qubit layout for the square-grid implementation of the gauging Clifford measurement. In the middle of each plaquette there is a flag qubit, that flags the three edges that are parallel with one of the boundaries. The top left corner is deformed to reduce the number of extra qubits (needed for the ancilla cycles) at the boundary. (b) Circuit schedule of the a gauging logical (Y) measurement step. The two detector slices show a contracting (exists in steps 1 to 8) and an expanding logical Y check (steps 2 to 15). (c) Circuit schedule for one stabilizer measurement round. Detector slices correspond to the contracting Z (blue) and the expanding X detectors (red). Contracting X and expanding Z detectors are not shown on the figure for clarity.
• Figure 3: (a) Gauging Clifford measurements on a growing the lattice (intermediate code distances $d_\text{int}\in\{3,5,7\}$). Starting with a unitary injection, one stabilizer measurement and one Clifford check is performed at every stage until the target distance, where the remaining $(d+1)/2$ rounds take place. (b) Cultivation in the strictly distance-preserving scheme with $d-1$ stabilizer rounds before every Clifford double check. Blue slices denote stabilizer measurements and imply growing with physical Bell-state preparations unless preceded with a Clifford check of the same distance.
• Figure 4: (a) Comparison of different simulation methods for the $d=3$ gauging logical measurement protocol under uniform circuit-level noise for various physical error rates $p$. The Clifford simulations of the $\ket{+i}$ state preparation, obtained via stim, align with that of the statevector simulation and the $\ket{T}$-state preparation. (b) Comparison between the gauging Clifford measurement and magic state cultivation for uniform circuit-level noise. For this noise model the break-even point in success probability is $d=9$. (c) Transitioning to a regular surface code patch in three rounds. The first round involves a reset, six CX, and a measurement layer; the second is similar with three CX rounds; the last one is a regular surface code syndrome cycle. See Fig. \ref{['fig:escape_full']} for the full circuit.
• Figure 5: (a) The same circuit as in the main text. (b) Pipelined circuit with 18 CX time steps and 3 measurement/reset time steps.