Experimental Demonstration of Logical Magic State Distillation

Abstract

Realizing universal fault-tolerant quantum computation is a key goal in quantum information science. By encoding quantum information into logical qubits utilizing quantum error correcting codes, physical errors can be detected and corrected, enabling substantial reduction in logical error rates. However, the set of logical operations that can be easily implemented on such encoded qubits is often constrained, necessitating the use of special resource states known as 'magic states' to implement universal, classically hard circuits. A key method to prepare high-fidelity magic states is to perform 'distillation', creating them from multiple lower fidelity inputs. Here we present the experimental realization of magic state distillation with logical qubits on a neutral-atom quantum computer. Our approach makes use of a dynamically reconfigurable architecture to encode and perform quantum operations on many logical qubits in parallel. We demonstrate the distillation of magic states encoded in d=3 and d=5 color codes, observing improvements of the logical fidelity of the output magic states compared to the input logical magic states. These experiments demonstrate a key building block of universal fault-tolerant quantum computation, and represent an important step towards large-scale logical quantum processors.

Figures (8)

• Figure 1: Logical magic state distillation factory.a, Schematic overview: Bloch sphere representation of magic state $\left|\psi_L\right>$ (left) pointing in the (1,1,1) direction with shaded region indicating noise. Distillation (center) takes multiple noisy logical inputs and produces a higher fidelity magic state (right). b, 5-to-1 distillation procedure (left to right). Non-fault-tolerant encoding of physical magic states into five data code logical qubits $\left|\psi_L\right>$ protects logical operations (i and ii). In particular, we encode into distance 3 and 5 color codes (i). Encoded states (ii) are purified using a distillation code. By running the un-encoding circuit of the distillation code (iii) and conditioning on distillation syndromes (iv), we have simultaneously projected into the code state of the distillation code and un-encoded it into the output magic state. Upon measuring the correct distillation syndromes, the output qubit has been "distilled" to a higher fidelity along the (1,1,1) direction. c, Averaged atom images from the $d\,{=}\,5$ distillation experiment, showing 85 physical qubits encoded into 5 logical qubits (LQ1 to LQ5) with 17 physical qubits each (left), shown here in SLM traps. Rows of logical qubits are coherently reconfigured for transversal CZ gates throughout the distillation circuit (right), shown here with LQ1 and LQ3 in AOD traps.
• Figure 1: Experimental layout of magic state distillation factory.a, We arrange 7 to 17 $^{87}$Rb atoms, each corresponding to a physical qubit, into a row. This horizontal register represents a logical qubit, tiled into 5 rows for a total of five logical qubits (LQ1 to LQ5). b, Encoding. Once the register of physical qubits is prepared, we coherently rearrange atoms to perform two-qubit entangling gates using the Rydberg blockade mechanism. We break up the circuit into "layers" each containing one set of local rotations, transport, and CZ gates. c, Coherent movement of logical qubits to perform transversal CZ gates. In the case of 5-to-1 distillation, this is achieved in three layers. The circuit as drawn here corresponds 1 to 1 to the atom layout, whereas in Fig. \ref{['fig:d3_distillation']} logical qubits LQ1 and LQ2 are swapped for clarity. d, Global measurement of qubits after circuit execution.
• Figure 2: Parallel logical encoding of arbitrary states.a, Circuit for injecting an arbitrary state $\left|\psi(\theta,\phi)\right>$ into the [[7,1,3]] color code. b, Schematic of $d\,{=}\,3$ color code stabilizers indicated by the three colored regions, with a logical operator highlighted. c, Bloch sphere representation of the injected state with varying angle $\phi$ on the XY plane (left). Error-corrected logical outcomes for X, Y, Z measurement basis versus the injected phase. Faded markers indicate outcomes upon postselection on perfect stabilizers. d, Left, Bloch sphere representation of the (1,1,1) magic state. Center, Injected $d\,{=}\,3$ magic state fidelity corresponding to raw, error-corrected and postselected on perfect stabilizers, averaged across all 10 logical qubits. Right, Spatial distribution of injected magic state fidelities.
• Figure 2: Experimental layout of $\mathbf{d\,{=}\,5}$ encoding. The arbitrary-state encoding circuit for the $d\,{=}\,5$ color code (left) is comprised of five entangling gate layers, illustrated by averaged images of the corresponding atom configurations (right), and local gates between the layers. We execute encoding with 5x parallelism, one instance per row (LQ1 to LQ5). The horizontal AOD trap array is tiled vertically by the second AOD. For each layer, atoms start in SLM sites, we apply local rotations, pick up and move atoms to their gate location, execute parallel CZ gates, echo (omitted for clarity), and finally move back to SLM sites.
• Figure 3: 5-to-1 magic state distillation.a, Magic state distillation circuit based on the [[5,1,3]] code (distillation code). We measure distillation syndromes in the Z basis and perform tomography on the distilled output. The successful distillation syndrome for this circuit is 1011 (Methods). b, Fidelity of the output magic state for the $d\,{=}\,3$ distillation (blue line for the MLE decoder, orange line for the MLD decoder, see main text) as a function of the total accepted fraction, which includes both sliding scale postselection on distillation syndrome stabilizers, and the factory acceptance (1/6 in the noiseless case). With sufficient stabilizer flagging, the output fidelity exceeds that of the input error-corrected magic state fidelity (green). The shaded regions indicate 68% confidence intervals, equivalent to $1\sigma$. c, We examine the distilled fidelity with full stabilizer postselection, after introducing coherent Z errors to the input magic states ($0.32\pi$, $0.24\pi$, $0.16\pi$ and $0$, left to right, blue points). The results are in good agreement with the theoretical expectation (gray line). The stars in b and c indicate the same data point. d, Factory acceptance rate of distillation syndromes after perfect stabilizer postselection with the same coherent errors as panel c. Dashed line indicates the $1/6$ acceptance rate of the 5-to-1 magic state factory in the noiseless case.