Universal logical operations in a silicon quantum processor - Nature

Download PDF AbstractQuantum errors induced by environmental noise are unavoidable and preclude the direct implementation of practical quantum computation. Fault-tolerant quantum computation offers one of the viable paths, necessitating the encoding and processing of information within logical qubits to curb such errors. Although substantial progress has been achieved recently in building silicon quantum computers, logical operations still haven’t been realized in silicon. Here we demonstrate a logical quantum processor using a phosphorus donor cluster in silicon. By implementing the [[4, 2, 2]] code, we realize the essential components for logical operations, which include fault-tolerant preparation of logical states and the characterization of a universal gate set comprising logical single-qubit and two-qubit gates. In particular, the logical T gate is achieved using the gate-by-measurement method, and magic states based on this gate are prepared. Furthermore, we execute the variational quantum eigensolver algorithm using two logical qubits and simulate the ground state of the electronic structure of the water molecule H2O. This work represents a key step towards scalable, fault-tolerant quantum computation in silicon spin qubits. Similar content being viewed by others Quantum logic with spin qubits crossing the surface code threshold Article Open access 19 January 2022 Tomography of entangling two-qubit logic operations in exchange-coupled donor electron spin qubits Article Open access 28 September 2024 Experimental fault-tolerant code switching Article 24 January 2025 MainSilicon-based spin qubits1,2 have emerged as a promising platform for scalable quantum computing, combining complementary metal–oxide–semiconductor (CMOS)-compatible fabrication technologies alongside the long quantum coherence times intrinsic to electron and nuclear spins in isotopically purified 28Si. Recent advances highlight the potential of silicon qubit platforms for scalable quantum processors, including quantum gate fidelities above the fault-tolerance threshold3,4,5,6, seminal attempts at quantum error correction7,8 and early-stage multi-qubit integration9,10,11,12. Additionally, initial endeavours to develop a quantum computing system-on-chip have also been undertaken, showcasing qubits operating above 1 K (refs. 13,14,15,16), cryo-CMOS chips rivalling the control precision of room-temperature instruments17 and a heterogeneous ‘chiplet style’ integrated quantum computing system-on-chip prototype18. As one of the major technical approaches in silicon, donor-based spin qubits occupy a unique niche, offering atomic-scale qubits and long coherence times19. Although challenges remain in atomic-scale device fabrication compared with gate-defined quantum dots, advances in donor systems now encompass high-fidelity quantum gate operations5, multi-qubit circuit demonstrations5,20, exchange-coupling-assisted multi-qubit gates between donor clusters12 and the exploitation of high-dimensional nuclear spin qudit states21—collectively positioning donor qubits as a compelling pathway towards a scalable, silicon-based quantum computing platform.Physical qubits in solid-state platforms1,22 are susceptible to environmental noise, which originates from multiple sources, including phonon interactions, nuclear spin fluctuations and charge noise. In silicon-based quantum processors, frequency crowding and cross-talk23,24,25,26 further exacerbate the errors as the system scales. To address these errors, logical encoding stands as the only viable solution by redundantly storing quantum information across multiple physical qubits27,28. While logical qubits and operations have been successfully demonstrated in platforms such as superconducting circuits29,30, neutral atoms31, nitrogen-vacancy centres32 and trapped ions33, their implementation in silicon-based spin qubits poses notable technical challenges. This is due to the need to simultaneously realize multi-qubit entanglement, deep circuit depth and multi-qubit measurements. Recently, two research groups have made pioneering advances towards fault-tolerant quantum computation (FTQC) by demonstrating phase error correction via coherent conditional rotations7,8. However, fault-tolerant (FT) encoding and universal logical operations have yet to be realized in silicon-based spin qubits.The [[4, 2, 2]] code34,35,36 has been highlighted by Gottesman for its minimal resource requirements in demonstrating quantum fault tolerance36. The code enables FT encoding of two logical qubits on only five physical qubits, with one serving as an ancillary that is not involved in the encoding36. Furthermore, it allows FT single-qubit and two-qubit Clifford logical gates, making the [[4, 2, 2]] code an ideal building block for scalable quantum computing architectures. As FTQC becomes mainstream in quantum computing research, the [[4, 2, 2]] code has become increasingly prominent31,37. Although this code cannot correct an arbitrary single-qubit error, it functions as the inner code in concatenated higher-level error-correction schemes38, which can substantially reduce resource overhead by orders of magnitude and lower the FT threshold in large-scale quantum computing39.Here we demonstrate a logical quantum processor based on the [[4, 2, 2]] code using five nuclear spins in a silicon donor system19. The FT logical state is prepared and a complete set of universal logical gates is demonstrated, including single-qubit and two-qubit gate operations, in a donor cluster. All Clifford gates are implemented using native physical quantum gates, while the T gate is realized through the gate-by-measurement method. Finally, we execute a variational quantum eigensolver (VQE) algorithm on two logical qubits using logical operations to compute the ground-state energy of a water molecule, H2O.Logical state preparationIn this work we employ scanning tunnelling microscopy (STM) lithography to fabricate our device on a Si(100)-2 × 1 surface. Following the lithography patterning, phosphorus atoms are doped into the opened windows and incorporated in subsequent processing. After the overgrowth of an epitaxial capping layer, the atomic device is embedded between the epitaxially grown 28Si layers. The device contains three donor cluster dots, though only the right one is used in this experiment, which consists of five phosphorus atoms (see Fig. 1a) and one fortuitously incorporated hydrogen atom. The average electron spin read-out fidelity for the right dot is estimated to be ~81.45%, and further details can be found in our previous work40. The five phosphorus nuclear spins are used for implementing the logical quantum circuits, while the hydrogen nuclear spin is initialized to a fixed spin-up state. The calibrated system parameters can be found in Supplementary Tables 1 and 2 of Supplementary Section 5. The state preparation and measurement (SPAM) error characterization and physical gate fidelity benchmarks are presented in Extended Data Figs. 1 and 2.Nuclear spins serve as qubits in our system. The native gate set comprises single-qubit gates realized through nuclear magnetic resonance (NMR) and CCCCZ-type gates implemented via electron spin resonance (ESR). Single-qubit addressability is enabled by distinct hyperfine coupling of each nuclear spin, which varies depending on their atomic configurations within the cluster. The CCCCZ-type gates are realized via driving the corresponding ESR transition of a specific nuclear spin configuration5,12,20. Consequently, an ESR 2π pulse induces a geometrical π phase, enabling electron-mediated nuclear spin CCCCZ-type gates. These gates offer high connectivity and high efficiency in quantum circuit compiling, which could be used to reduce circuit complexity41. However, some ESR driving frequencies could be closely spaced due to near-degenerate hyperfine couplings, necessitating cross-talk suppression techniques (detailed in Supplementary Section 11).To prepare the logical two-qubit states, we compile the logical state preparation circuits for logical state \({\left|00\right\rangle }_{{\rm{L}}}\) and Bell state \({\left|{\Phi }^{+}\right\rangle }_{{\rm{L}}}\) using our native gate set. Both circuits use four physical qubits, with their compiled implementations and error propagation analysis detailed in Supplementary Section 6. The prepared states \({\left|00\right\rangle }_{{\rm{L}}}\) and \({\left|{\Phi }^{+}\right\rangle }_{{\rm{L}}}\) are measured in the physical qubit basis (the corresponding expectation values under different bases for \({\left|00\right\rangle }_{{\rm{L}}}\) state reconstruction are provided in Extended Data Fig. 3), and their corresponding state tomography results are shown in Fig. 1b,c, yielding initial physical fidelities of 84.2(5)% and 80.5(7)%, respectively (the representation of the uncertainty is explained in Methods).Fig. 1: The donor cluster and preparation of the logical states.Full size imagea, The schematic of a donor cluster with five 31P nuclei in a 28Si lattice. As detailed in our previous work40, the hyperfine interactions (A) between the shared electron (light blue background) and each nucleus (marked in different colours) are A1 = 28.6 MHz, A2 = 73.7 MHz, A3 = 137.0 MHz, A4 = 226.0 kHz and A5 = 168.0 kHz. Those hyperfine interactions enable high-connectivity CCCCZ-type gates within one cluster. b,c, Experimental reconstructed density matrices ρ for the logical state \({\left|00\right\rangle }_{{\rm{L}}}\) (b) and the logical Bell state \({\left|{\Phi }^{+}\right\rangle }_{{\rm{L}}}\) (c), obtained via quantum state tomography (Methods). d, The infidelity of the logical \({\left|00\right\rangle }_{{\rm{L}}}\) and Bell \({\left|{\Phi }^{+}\right\rangle }_{{\rm{L}}}\) states, for the raw data, after \({\widehat{S}}^{Z}\) PP, \({\widehat{S}}^{X}\) PP and both of them, where PP is the parity projection. Compared with the raw data, the error can be suppressed over four times. The dashed wireframes in d depict results from master equation simulation (Sim). e,f, Postprocessed logical density matrices ρL of logical \({\left|00\right\rangle }_{{\rm{L}}}\) (in e) and logical \({\left|{\Phi }^{+}\right\rangle }_{{\rm{L}}}\) (in f) after the PP on both \({\widehat{S}}^{Z}\) and \({\widehat{S}}^{X}\) stabilizers. The wireframes in b, c, e and f represent ideal results. In b–f, each basis for quantum state tomography comprises at least 300 measurement repetitions. All error bars are estimated via Monte Carlo bootstrap resampling and represent one standard deviation (1 s.d.) from the mean.Source dataThe [[4, 2, 2]] code has stabilizers \({\widehat{S}}^{X}={X}_{1}{X}_{2}{X}_{3}{X}_{4}\) and \({\widehat{S}}^{Z}={Z}_{1}{Z}_{2}{Z}_{3}{Z}_{4}\) (X, Y and Z denote the corresponding Pauli operators across this paper). Since we cannot perform mid-circuit measurements, stabilizer parity projections are implemented via postprocessing at the end of circuits in a destructive manner. After postprocessing with \({\widehat{S}}^{Z}\) and \({\widehat{S}}^{X}\) parity projections and removing detected errors, the state fidelities of \({\left|00\right\rangle }_{{\rm{L}}}\) and \({\left|{\Phi }^{+}\right\rangle }_{{\rm{L}}}\) in the logical space increased to 96.5(6)% and 95.5(9)%, as shown in Fig. 1d, with acceptance ratios of 83.9(13)% and 82.9(17)%, respectively. Their converted state density matrices on the logical basis are depicted in Fig. 1e,f. Error analysis reveals that the logical error originates mainly from cross-talk-induced higher-weight errors, which lie beyond the error detection capability of the [[4, 2, 2]] code. In addition, we prepared the logical state \({\left|00\right\rangle }_{{\rm{L}}}\) in a circuit that has a similar spirit, proposed by Gottesman with a flag qubit36, achieving a logical fidelity of 97.3(7)% after postprocessing with both stabilizers conditioned on the flag not being raised (Extended Data Fig. 4).Next we characterize the logical coherence times by measuring the expectation values of the logical operators XLIL, ILXL and XLXL as a function of delay time τ (ref. 42). The results for different logical states are displayed in Fig. 2a, and by fitting with the approximately exponential decay, we obtain logical coherence times as \({\tau }_{{X}_{{\rm{L}}}{I}_{{\rm{L}}}}=230(9)\,{\upmu}{\rm{s}}\), \({\tau }_{{I}_{{\rm{L}}}{X}_{{\rm{L}}}}=140(12)\,{{\upmu }}{\rm{s}}\) and \({\tau }_{{X}_{{\rm{L}}}{X}_{{\rm{L}}}}=255(12)\,{{\upmu }}{\rm{s}}\), which are related to the logical qubit L1 with \(\left|+0 \right\rangle _{{\rm{L}}}\), logical qubit L2 with \({\left|0+\right\rangle }_{{\rm{L}}}\) and their entangled state with \({\left|{\Phi }^{+}\right\rangle }_{{\rm{L}}}\), respectively. The average coherence time for physical qubits (except N4 and N5) is approximately 523 μs, while the average coherence time for logical qubits is around 208 μs, which is shorter than that of all the physical qubits. This reduction arises because the two logical qubits are, by their nature, a four-qubit entangled state that is inherently more susceptible to decoherence compared with any individual physical qubit. Notably, the measurement results in Fig. 2b reveal a strong noise bias in the system, with phase-flip (Z) errors dominating bit-flip (X) errors. Based on previous theoretical works, the biased noise can be beneficial in improving the fault-tolerance threshold, thereby reducing the physical qubit overhead and simplifying scalable architectures43.Fig. 2: Logical coherence times and error rates.Full size imagea, The logical coherence times and lifetimes are characterized, where the expectation values 〈XLIL〉 and 〈ILZL〉 are measured on \({\left|+0\right\rangle }_{{\rm{L}}}\), 〈ILXL〉 and 〈ZLIL〉 on \({\left|0+\right\rangle }_{{\rm{L}}}\) and 〈XLXL〉 and 〈ZLZL〉 on \({\left|{\Phi }^{+}\right\rangle }_{{\rm{L}}}\). Since our qubits are encoded on nuclear spins, the expectation values of ZLIL, ILZL and ZLZL remain stable without observable decay throughout the measurement duration, reflecting their inherently long lifetimes. b, The error rates are obtained through postprocessing of stabilizer parity checks (Supplementary Section 8 contains details). The qubit flip error (X), detected by the stabilizer \({\widehat{S}}^{Z}\), remains at its initial value. By contrast, the phase-flip error (Z), detected by the stabilizer \({\widehat{S}}^{X}\), increases as the physical qubits decohere versus delay time t. Each data point is averaged over 1,000 experimental repetitions. Error bars represent 1 s.d. from the mean, estimated via bootstrap resampling.Source dataUniversal logical gate setWe implement a universal logical gate set, comprising single-qubit gates XL, SL and TL, the simultaneous single-qubit gate HLHL and the two-qubit gate CNOTL. Since the circuits for demonstrating logical single-qubit gates on either logical qubit are similar, here we perform those operations on the first logical qubit as an example, labelled as XLIL, SLIL and TLIL. The implementation circuits for those logical Clifford gates are listed in Fig. 3a, where the physical single-qubit operations are realized via NMR pulses, the physical CZ gate is executed by the corresponding CCCCZ gate and SWAP gates in the HLHL and CNOTL circuits are virtually applied through qubit relabelling (Supplementary Section 15 contains the compiled circuits). Notably, in the compiled circuits, the gates XLIL, HLHL and CNOTL are FT, whereas the SLIL gate is non-FT (nFT).Fig. 3: Logical gates for the [[4, 2, 2]] code.Full size imagea, Circuits for the logical Clifford gates of the [[4, 2, 2]] code, comprising XLIL, SLIL, HLHL and CNOTL. The corresponding logical circuits are shown in the middle column, and their equivalent physical circuits are displayed in the last column. b, Measured population transfer fidelities of logical gates XLIL, CNOTL and HLHL, where the red dashed bars represent the simulation results. c, The QPT result for the SLIL gate on the first logical qubit. For different input states, the resulting density matrices are measured after applying the SLIL gate, achieving a QPT fidelity of 87.0(16)%. The wireframes depict the ideal results.Source dataThe FT logical gates XLIL, HLHL and CNOTL are characterized by their population transfer fidelity \({F}_{{\rm{L}}}={\rm{Tr}}({M}_{\exp }{M}_{{\rm{ideal}}}^{-1})\) in the logical space, where Mexp and Mideal represent the experimental and ideal population transfer matrices in the logical basis, respectively. For XLIL and CNOTL, the initial states are \({\left|00\right\rangle }_{{\rm{L}}}\), \({\left|01\right\rangle }_{{\rm{L}}}\), \({\left|10\right\rangle }_{{\rm{L}}}\) and \({\left|11\right\rangle }_{{\rm{L}}}\); for HLHL, the initial states are \({\left|++\right\rangle }_{{\rm{L}}}\), \({\left|+-\right\rangle }_{{\rm{L}}}\), \({\left|-+\right\rangle }_{{\rm{L}}}\) and \({\left|--\right\rangle }_{{\rm{L}}}\). The resulting state transfer matrices Mexp can be found in Extended Data Fig. 5, and the corresponding population transfer fidelities for XLIL, CNOTL and HLHL are 88.3(5)%, 88.6(4)% and 75.6(4)%, respectively. The average fidelity of both physical single-qubit and two-qubit gates exceeds 95%, whereas the average fidelity for logical single-qubit and two-qubit gates is approximately 86%. This discrepancy arises because the error from the initial logical state preparation circuits was not subtracted during the characterization of the logical gates (Fig. 3b; Supplementary Section 16 has details).The SLIL gate essentially realizes a π/2-phase σz rotation on the first logical qubit. We used the gate process tomography to calibrate the SLIL gate, with \({\left|00\right\rangle }_{{\rm{L}}}\), \({\left|10\right\rangle }_{{\rm{L}}}\), \({\left|+0\right\rangle }_{{\rm{L}}}\) and \({\left|+i0\right\rangle }_{{\rm{L}}}\) as the initial input states. The italic i denotes the imaginary unit. In the state definition: |+i〉 = (|0〉 + i|1〉) / √2; this represents the eigenstate of the Pauli-Y operator, corresponding to the +Y axis on the Bloch sphere. Figure 3c presents the quantum process tomography (QPT) results (details shown in Extended Data Fig. 6), revealing an 87.0(16)% gate process fidelity. Across all the logical gates, the experimental results align well with our master equation simulations, with residual cross-talk identified as the dominant error source.While Clifford gates are essential, universal quantum computing necessitates the inclusion of non-Clifford gates; otherwise, the computation can be efficiently simulated classically, as per the Gottesman–Knill theorem44. To complete our universal logical gate set, we implement a logical TLIL gate using the gate-by-measurement method30, which is crucial for achieving FTQC. As shown in Fig. 4a, the circuit employs five nuclear spins, using an ancillary nuclear spin (Qa) to inject the TLIL and \({T}_{{\rm{L}}}^{\dagger }{I}_{{\rm{L}}}\) gate onto the target logical qubit (L1) via controlled operations. Specifically, the ancillary read-outs \(\left|0\right\rangle\) and \(\left|1\right\rangle\) correspond to a TLIL and \({T}_{{\rm{L}}}^{\dagger }{I}_{{\rm{L}}}\) gate for a π/4 and –π/4 Z rotation, respectively. Here we postselect the ancillary \(\left|0\right\rangle\) outcomes to evaluate the TLIL gate, where the QPT reveals a fidelity of 82.6(23)% (Fig. 4b). Meanwhile, the \({T}_{{\rm{L}}}^{\dagger }{I}_{{\rm{L}}}\) gate QPT fidelity is measured as 91.5(24)% (Extended Data Fig. 7). We further verify the functionality of the TLIL gate by applying it to different input logical cardinal states and performing output state tomography, with results shown in Fig. 4c. In addition, Extended Data Table 1 provides all corresponding logical state characterization results.Fig. 4: QPT for the TLIL gate.Full size imagea, Circuit for QPT of the TLIL on the first logical qubit L1, where ‘State prep.’ denotes state preparation and the four logical cardinal states {\({\left|00\right\rangle }_{{\rm{L}}},{\left|10\right\rangle }_{{\rm{L}}},{\left|+0\right\rangle }_{{\rm{L}}},{\left|+i0\right\rangle }_{{\rm{L}}}\)} are prepared as inputs. The logical gate is implemented via the gate-by-measurement method using an extra ancillary nuclear spin. b, Experimental (solid) and ideal (wireframe) QPT matrices conditioned on ancillary measurement outcome \(\left|0\right\rangle\), giving a TLIL gate process fidelity of 82.6(23)%. c, Logical state tomography of input and output states of the TLIL gate. The inset Bloch spheres display the initial states (blue dots) and final states (green dots) under the TLIL gate operation in the logical code space. The wireframes depict ideal values.Source dataMagic states are indispensable resources for achieving universal FT gate sets, and high-fidelity magic states can be purified through distillation45. Notably, the TLIL and \({T}_{{\rm{L}}}^{\dagger }{I}_{{\rm{L}}}\) gates we implement directly generate H-type equivalent magic states when applied to \({\left|+0\right\rangle }_{{\rm{L}}}\) and \({\left|+i0\right\rangle }_{{\rm{L}}}\). The resulting states are defined as follows: \({\left|H\right\rangle }_{1}={T}_{{\rm{L}}}{I}_{{\rm{L}}}{\left|+0\right\rangle }_{{\rm{L}}}\), \({\left|H\right\rangle }_{2}={T}_{{\rm{L}}}^{\dagger }{I}_{{\rm{L}}}{\left|+0\right\rangle }_{{\rm{L}}}\), \({\left|H\right\rangle }_{3}={T}_{{\rm{L}}}{I}_{{\rm{L}}}{\left|+i0\right\rangle }_{{\rm{L}}}\) and \({\left|H\right\rangle }_{4}={T}_{{\rm{L}}}^{\dagger }{I}_{{\rm{L}}}{\left|+i0\right\rangle }_{{\rm{L}}}\), and their corresponding state tomography reveals magic-state fidelities of 89.8(21)%, 89.7(23)%, 84.0(23)% and 95.2(19)%, respectively. The fidelities of magic state \({\left|H\right\rangle }_{4}\) surpass the distillation threshold of 92.7% for H-type states, which enables the universal quantum computation through the adaptive magic-state distillation protocol proposed by Bravyi and Kitaev45. Furthermore, we quantify the amount of magic using stabilizer 2-Rényi entropy46, for which the theoretical maximum is 0.415 for H-type magic states and 0.585 for T-type magic states. Our \({\left|H\right\rangle }_{1}\), \({\left|H\right\rangle }_{2}\), \({\left|H\right\rangle }_{3}\) and \({\left|H\right\rangle }_{4}\) states yield magic values of 0.445(15), 0.420(32), 0.402(22) and 0.445(13), respectively, confirming the magic produced by our quantum circuit. The slight deviations above the theoretical maximum of 0.415 arise from imperfections in our H-type magic-state preparation, causing the states to deviate from the equatorial plane of the logical Bloch sphere (Supplementary Section 9 contains details).Variational quantum eigensolver demonstrationFinally, we demonstrate the practical utility of the logical qubit architecture by executing a VQE algorithm47. Using two logical qubits encoded in the [[4, 2, 2]] code, we compute the ground-state energy of the electronic structure of the water molecule48 through the optimization of the bond angle φ between the O–H bonds with a fixed bond length r = 1.81 Å. We implement the unitary coupled cluster ansatz \({\left|\psi (\theta )\right\rangle }_{{\rm{L}}}={{\rm{e}}}^{-i\theta {Y}_{{\rm{L}}}{X}_{{\rm{L}}}/2}\left|\Phi \right\rangle\), where \(\left|\Phi \right\rangle ={\left|10\right\rangle }_{{\rm{L}}}\) represents the Hartree–Fock reference state and θ is the variational parameter to be optimized. By considering an active space of two electrons in two active orbitals (Supplementary Section 10 for details), the water Hamiltonian under the Bravyi–Kitaev transformation can be written as H = g0 + g1ZLIL + g2ILZL + g3ZLZL + g4XLXL + g5YLYL, where {gi} are the coefficients that depend on the bond angle φ. As illustrated in Fig. 5a, each workflow iteration comprises both quantum and classical stages, where the quantum stage prepares the parameterized state \({\left|\psi (\theta )\right\rangle }_{{\rm{L}}}\) and performs measurements, and the classical stage updates the parameter via postprocessing steps. Then, we feed the updated θ into the next iteration cycle, until convergence is achieved. Ultimately, the potential-energy curve is obtained by repeating the convergence process while sweeping different bond angles φ.Fig. 5: H2O ground-state energy computation by VQE.Full size imagea, VQE workflow schematic, which consists of the quantum and classical stages. The quantum circuit prepares logical state \({\left|10\right\rangle }_{{\rm{L}}}\), applies the unitary coupled cluster ansatz for trial state \({\left|\psi (\theta )\right\rangle }_{{\rm{L}}}\) and performs partial tomography (Rt), while the classical stage processes expectation values via Clifford fitting (CF) and symmetry verification (SV) for error mitigation (Methods). The classical optimizer (Nelder–Mead algorithm) iteratively updates parameter θ until energy convergence. b, H2O potential-energy surface versus bond angle φ. Exact Hamiltonian diagonalization (red) is compared with