Teleportation transition of surface codes on a superconducting quantum processor

Abstract

The topological surface code is a leading candidate for harnessing long-range entanglement to protect logical quantum information against errors, and teleportation of logical states is desirable for robust quantum information processing. Nevertheless, scaling up the surface code in quantum teleportation poses a formidable challenge to experiment. Here on a superconducting quantum processor with 125 qubits, we demonstrate the robust teleportation of topological rotated surface code prepared by a linear-depth unitary circuit, with code distances up to 7. We obtain the teleportation phase diagram by tuning the local entangling gates uniformly across a finite threshold. Furthermore, we show that the entangling threshold can be boosted by coherent qubit rotations that inject magic resources beyond the Clifford regime, restoring the duality symmetry of the topological phase, which serves as a guiding principle to minimize the entanglement resource. Our results shed light on simulating and leveraging topological quantum matter on quantum devices, and pave the way to the ultimate goal of distributed fault tolerant quantum computation.

Figures (6)

• Figure 1: Experimental setup and conceptual illustration. a, Our 125-qubit quantum processor and two embedded distance-7 surface codes, labelled Alice (purple) and Bob (green). Each surface code comprises 49 qubits, with the logical subspace defined by $Z$-type stabilizers and $X$-type stabilizers. In the subsequent experiments, logical states are teleported from Alice to Bob, as indicated by the light blue arrow. Alice and Bob are physically interlaced to facilitate this state teleportation. b, Schematic illustration of the tunable teleportation protocol. The process begins with the initialization of Alice in an encoded logical state, while Bob's qubits are prepared in the physical product state $\ket{0\cdots0}$. A parameterized coupling operation, defined by parameters $\theta$ and $t$, is then applied between the systems to induce entanglement. The measurement outcome of Alice is then communicated to Bob, upon which local corrections are applied. The final logical information is retrieved via a classical decoding algorithm. c, Topological phase diagram under coherent errors in the $XZ$-plane. When the injected coherent error magnitude $t$ is small (shaded blue region), logical information can be faithfully recovered after imperfect teleportation, demonstrating that the topological order of the surface code is preserved. Rotating the coherent error axis from the $X$ direction (orange arrow) to the $X+Z$ direction (red arrow), significantly enhances the entangling threshold. The solid lines indicate the entangling threshold for teleportation, while the dashed line indicates the learning threshold of Alice. d, Phase diagram under incoherent bit-flip noise. In addition to coherent errors, we analyse the performance of logical teleportation in the presence of incoherent noise, modelled here by an independent bit-flip channel acting on the physical qubits. The teleportation of topological logical state becomes infeasible when the noise strength surpasses the critical threshold $p_\mathrm{c}=0.109$.
• Figure 2: Circuit and stabilizer parities before and after teleportation.a, Schematic circuit for state preparation and teleportation for code distance $2$. Initializing from a product state, after the first stage Alice's qubits (white circles with even numbers) are prepared in the logical state $\ket{0_\mathrm{L}}$ of a surface code, while Bob's qubits (white circles with odd numbers) stay in the product state $\ket{0\cdots0}$. In the second stage, the unitary gates entangle Alice's and Bob's qubits one-on-one (i.e., transversally). After measuring out the qubits of Alice, the state of surface code is successfully transferred from Alice to Bob i.e., teleportation when involving only two parties. The gate parameter $t$ plays the role of "coherent error", that tunes the entangling, between Alice and Bob, from maximal ($t=0$) to minimal ($t=\pi/4$). Here the feedback correction of the measurement outcomes are performed in advanced by the CNOT gates before the measurement of Alice's qubits, such that after these CNOT layers, Alice and Bob are effectively disentangled when $t=0$, as shown in the schematic. The rotation angle $\theta$ is a knob that does not play a role when the entangling is maximal $t=0$, but it can tune the threshold of coherent error, as will be shown later. b, Measured $X$-type and $Z$-type stabilizer parity values for the logical state $\ket{0_\mathrm{L}}$ using Alice's qubits (left) and Bob's qubits (right). To calculate the $X$ and $Z$ parities, we simultaneously measure all of Alice's (or Bob's) qubits in the $X$ and $Z$ basis, respectively, over $4\times10^4$ measurement shots, followed by readout error mitigation (SI Section S1C). Mean parity: $0.82(8)$ for Alice and $0.68(8)$ for Bob. Using the ML decoder, we estimate the expectation value of the logical operator $Z_\mathrm{L}$ to be $0.973(2)$ for Alice and $0.949(6)$ for Bob. The corresponding logical error rate is given by $\left(1 - \left\langle Z_\mathrm{L} \right\rangle\right)/2$.
• Figure 3: Teleportation transition by tuning entanglement.a, Logical $X$ error rate as a function of the effective entangling parameter $t_\mathrm{eff}$ for errors injected along the $X$ axis ($\theta=\pi/2$). The parameter $t_\mathrm{eff}$ combines both coherent and incoherent errors (see the main text). Experimental results (circles with error bars) fit to a pseudo-threshold of $0.122(2)\pi$ (vertical grey solid line) that is slightly higher than the asymptotic threshold of $0.107\pi$ (purple star). The pseudo-threshold is extracted from a finite-size scaling analysis shown in the inset. Solid colored lines: noiseless simulation results. b, Logical $X$ error rate as a function of the effective entangling parameter $t_\mathrm{eff}$ for errors injected along the $X+Z$ axis ($\theta=\pi/4$). At this angle, the system exhibits an electric-magnetic duality, leading to a significant enhancement of the entangling threshold. The experimentally observed pseudo-threshold is $0.175(5)\pi$, while the asymptotic threshold is $0.155\pi$.
• Figure 4: Teleportation of arbitrary logical states.a, Logical measurement results before teleportation (Alice, $d = 3$). These states are chosen to span the $XZ$-plane of the logical Bloch sphere, parameterized by the logical polar angle $\theta_{\rm{L}}$ (left panel). The right panel shows the measured expectation values of the logical operators $X_{\rm{L}}$ and $Z_{\rm{L}}$ for these states. Solid lines are fitting results, and dashed lines indicate noiseless ideal results. The $H$-type magic state at $\theta_{\rm{L}}=\pi/4$ is highlighted by the green dashed box. Average fidelity: $0.96(1)$. b, Logical measurement results after teleportation (Bob, $d = 3$). The average state fidelity of Bob's logical qubit is $0.95(2)$, only slightly lower than that of the initial states on Alice, reflecting a high-fidelity teleportation process. In all plots, error bars are significantly smaller than the corresponding marker size.
• Figure 5: Cumulative distributions of the operation errors for the distance-7 teleportation experiment. Red: Pauli errors for simultaneous single-qubit gates. Black: Pauli errors for simultaneous CZ gates. Blue: identification errors for state discriminations. The vertical dashed lines indicate median fidelities: $99.94\%$ for single-qubit gates, $99.64\%$ for two-qubit gates, and $99.08\%$ for readout.