Teleportation transition of surface codes on a superconducting quantum processor

The gist The authors experimentally demonstrate a teleportation protocol for a logical qubit of the surface code with imperfect entanglement resource, charting out the many-body teleportation transition across an entangling threshold by tuning local entangling gates and coherent qubit rotations.

Quantum Error Correction Context

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. It leverages topological order to robustly protect quantum information against errors, where errors are detected by measuring two types of stabilizers called syndromes, and are corrected by corresponding Pauli operations. Scaling up the surface code is challenging but necessary to achieve higher tolerance against the errors. The scaling also poses a challenge for the many-body generalization of the quantum teleportation, which is an exotic quantum phenomenon that harnesses entanglement resource to enable quantum communication between two systems.

Experimental Setup and Protocol

The experiment utilizes a high-performance 125-qubit superconducting quantum processor featuring tunable nearest neighbour couplings. The setup consists of two groups of qubits, referred to as Alice and Bob, each arranged in a d × d square lattice. A surface code is prepared on Alice prior to teleportation, while Bob remains in the product state 0 · · · 0⟩. The protocol involves applying a parameterized coupling operation defined by parameters θ and t between the systems to induce entanglement. The process begins with the initialization of Alice in an encoded logical state, while Bob’s qubits are prepared in the physical product state 0 · · · 0⟩.

Teleportation Transition Analysis

The researchers obtained a teleportation phase diagram by tuning local entangling gates uniformly across a finite threshold. They showed 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. The teleportation transition is charted by sweeping coherent error strength in their protocol to validate a stable and long-range phase. When 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.

Phase Diagram and Threshold Enhancement

The analysis shows that rotating the coherent error axis from the X direction to the X + Z direction significantly enhances the entangling threshold. This rotation injects magic resources beyond the Clifford class, which is optimal at angle θ = π/4. The effective entangling parameter teff is defined by sin2(teff) = sin2(t) + p (1 + cos(2t)) in the self-dual basis. This leads to an enhanced threshold, such as tc, exp (θ = π/4) = 0.175(5)π. The teleportation threshold is mathematically equivalent to the phase transition of the underlying quantum state of matter.

Decoding and Error Mitigation

The study compares different decoding methods, including tensor network (TN)-based maximum likelihood (ML) decoder, minimum-weight perfect matching (MWPM) decoder, and TN-based Bayesian decoder. The ML decoder yielded a logical error rate of 0.014(1) for Alice and 0.025(3) for Bob under teleportation below the threshold. The MWPM decoder extracted thresholds of 0.116(2)π at θ = π/2 and 0.164(2)π at θ = π/4, which appear slightly overestimated compared to the asymptotic theoretical thresholds.

Conclusion and Outlook

The work provides a comprehensive experimental characterization of logical state teleportation and establishes a foundational toolkit for probing the resilience of encoded information against tailored noise. The results demonstrate that electric-magnetic duality enhances the entangling threshold, suggesting that fostering competition between electric and magnetic excitations can enhance protection against errors. Future directions include incorporating fast mid-circuit measurement protocols and extending the protocol to a truly nonlocal setting between two superconducting processors.

Supplementary Information Details

The supplementary information details the effective incoherent noise strength p, which is modeled as a linear function p(d) = a(d + d0), where dmax is approximately 17 for the distance-7 code. The readout error mitigation procedure involves using a tensor-product noise model to accurately describe measurement errors. The analysis of Rényi relative entropy, D2(ρ0ρ1), maps the problem to a classical 2D Ising model, yielding a critical point at qc = 0.178. The circuit for preparing both Alice and Bob in a logical state involves SWAP gates to dynamically transfer qubits between the two lattices. The final set of generators for the distance-3 surface code is explicitly listed.

Device Performance Metrics

The device performance includes median fidelities for single-qubit and two-qubit gates of 99.94% and 99.64%, respectively. The median readout fidelity achieved is 99.08% by pumping each qubit from the state 1⟩ to the state 2⟩ prior to the readout pulse. The energy relaxation time T1 and spin-echo dephasing time TSE2 have median values of about 70.1 µs and 18.2 µs, respectively. The effective bit-flip error strength p is modeled as p = (1 − p0) pflip + (1 − pflip)p0 in the classical simulation. The finite size scaling analysis extracts a threshold of tc, exp (θ = π/2) = 0.122(2)π. The asymptotic thresholds for coherent and incoherent errors are pc = 0.109 and tc (θ = π/4) = 0.1548014(3)π. The final set of generators is equivalent to the stabilizer group defining the logical state 0L⟩ of the distance-3 surface code. The representative qubits and CNOT sequence are chosen to enable parallel gate implementation and minimize the total circuit depth. The number of two-qubit gate layers scales linearly as 3 ⌊(d + 3) /2⌋.

State Preparation Circuit Details

The circuit for preparing Alice in a logical state 0L⟩ begins with physical qubits initialized in the product state 0 · · · 0⟩. This is stabilized by single-qubit Pauli operators, and the encoding circuit evolves this initial state into a simultaneous eigenstate of all stabilizers with an eigenvalue of +1, including the X-type ones. The circuit for preparing both Alice and Bob in a logical state 0L⟩ involves SWAP gates to dynamically transfer qubits between the two lattices. This systematic process ultimately prepares Alice and Bob in the logical product state 0L⟩ ⊗ 0L⟩.

Decoding Methods Comparison

The MWPM decoder is well-known to find the shortest path to pair up the syndromes which can be efficiently found by algorithm like Blossom algorithm. The TN-based Bayesian decoder outputs a probabilistic correction that minimizes the “free energy” of the statistical model by taking into account the “entropy” as a key factor. The MWPM decoder yields thresholds of 0.116(2)π at θ = π/2 and 0.164(2)π at θ = π/4, which appear slightly overestimated compared to the asymptotic theoretical thresholds. The TN-based Bayesian decoder yields thresholds of 0.107(1)π at θ = π/2 and 0.152(2)π at θ = π/4, which agree very closely with the asymptotic theoretical thresholds.

Logical State Fidelity

The logical state fidelity FL, given by FL = 1/2 (1 + cos θL⟨ZL⟩ + sin θL⟨XL⟩), is shown in Fig. 4a. The experimental results are found to be captured by the functions ⟨XL⟩ = (1 − 2pZ) sin(θL) and ⟨ZL⟩ = (1 − 2pX) cos(θL), which fit two asymmetric logical error probabilities: pX = 0.019(1) and pZ = 0.053(1). The logical state fidelity FL is also shown in Fig. 4a.

Error Asymmetry

The error asymmetry is further confirmed using the H-type magic state, where the ideal expectation values of ⟨XL⟩ and ⟨ZL⟩ are equal. Instead, our experiment yields a lower ⟨XL⟩ = 0.62(1) compared to ⟨ZL⟩ = 0.68(1) <ref:

1. Increased fault tolerance in quantum computation by using surface codes for logical qubit protection against errors. This allows for more robust quantum information processing compared to less protected methods, as it leverages topological order to robustly protect quantum information against errors.

2. Robust teleportation of logical states across a 125-qubit superconducting processor with code distances up to 7. The system can perform this task even with imperfect entanglement resource by tuning entangling gates continuously, charting out the many-body teleportation transition across the entangling threshold.

3. Enhanced entangling threshold by utilizing coherent qubit rotations that inject magic resources beyond the Clifford regime. This technique restores the duality symmetry of the topological phase, which serves as a guiding principle to minimize entanglement resource during teleportation.

4. Development of an adaptive noise suppression protocol for quantum circuits using mid-circuit measurements and qubit resets. This enables Alice and Bob to perform adaptive circuits to suppress incoherent noise before and after the teleportation process, also allowing classical communication between Alice and Bob.

5. Improved logical state preparation fidelity by employing stabilizer measurement-based protocols instead of unitary encoding. This approach results in slightly lower logical error rates for all states compared to unitary methods, demonstrating a significant alleviation of error asymmetry.

6. Robust decoding of noisy quantum codes using both Tensor Network based maximum likelihood and Minimum Weight Perfect Matching decoders. These decoders allow the system to accurately infer the logical state from measured syndromes, yielding thresholds that agree very closely with the asymptotic theoretical thresholds.

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.