Scientists achieved quantum error correction of diamond quantum memory

Scientists believe quantum computers can run thousands of times faster than conventional computers and have the potential to be a game-changing future technology in fields such as chemistry, cryptography, finance, pharmaceuticals, and more. To harness this power, scientists are looking for ways to build networks of quantum computers; where fault-tolerant quantum memory, which responds well to hardware or software failures, will play an important role in building networks.

Recently, a research group from Yokohama National University demonstrated quantum error correction in a quantum memory using a diamond nitrogen vacancy (NV) center with two nuclear spins of surrounding carbon isotopes; using three-dimensional coils to cancel residual magnetic fields including the geomagnetic field , to achieve zero magnetic field; and tested three-qubit error correction for bit-flip or phase-flip errors at zero magnetic field. Quantum error correction enables quantum memory to withstand operational or environmental errors without the need for magnetic fields, opening the way for distributed quantum computing and a quantum internet with memory-based quantum interfaces or quantum repeaters.

For the first time, the research team demonstrated the quantum operation of electrons and nuclear spins in the absence of a magnetic field. The related research results were published in the journal "Communications-Physics" [1].

Nitrogen-vacancy (NV) centers in diamond are used as quantum memory, with error-correcting coding to automatically correct errors.

"Quantum error correction makes quantum memory resilient to operational or environmental errors without the need for a magnetic field, and opens the way for distributed quantum computing and a quantum internet with memory-based quantum interfaces or quantum repeaters." Yokohama University professor Hideo Kosaka, lead author of the study, said the team plans to take the technology a step further: "We hope to develop a quantum interface between superconducting and photonic qubits to enable fault-tolerant large-scale quantum computers."

Introduction to the experimental system

Under a magnetic field, increasing the fidelity of the controlled phase (C-Z) gate by adding an external magnetic field strongly depends on the carbon position relative to the NV center, unless the field is well aligned with the hyperfine magnetic field (Fig. ab,c); In contrast, the fidelity can be increased independently of the carbon position at zero magnetic field, where the hyperfine field uniquely defines the quantization axis, which allows time-reversal operations at any timing, thus Generates high-fidelity single-nuclear spin manipulation regulated by NV electrons. The application of external magnetic fields also limits integration with other physical qubits. For example, superconducting qubits become unstable by applying a magnetic field due to the penetration of magnetic flux into the superconductor or Josephson junction.

In this study, the team first demonstrated the quantum manipulation of electron and nuclear spins and their related operations based on geometric phases in the absence of a magnetic field; then used three nuclear spins on nitrogen and two carbon isotopes near the NV center. Spin, demonstrating the most basic three-qubit quantum error correction (Quantum Error Correction, QEC) for bit flip or phase flip errors.

Figure 1: Carbon spin manipulation with and without a magnetic field. a) Schematic structure of a nitrogen vacancy center in diamond, where the carbon (C) adjacent to the vacancy (e) is replaced by impurity nitrogen (N), whose spin error is protected by two carbon isotope spins for quantum error correction the South Pole qubit. b) The quantized axis of the carbon nucleus spin, determined by the ultrafine field of the electron spin and the applied external magnetic field. c) Based on two carbon nucleus spins at zero magnetic field (blue line), 100 Gauss (green line), 1000 Gauss (red line) and 10000 Gauss (orange line). For both carbon nuclear spins, the angle between the external magnetic field and the hyperfine magnetic field is set to 45 degrees. Under a magnetic field, the fidelity is limited by the influence of the two quantum axes, but at zero magnetic field, the fidelity is improved by reducing the Rabi frequency. The nitrogen nuclear spin is not considered here because the quantization axis does not change with the presence of a magnetic field. d) Energy level diagram of the correlated spin system containing one nitrogen and two carbon isotopes.

Three-qubit entanglement

The researchers first assessed the microwave intensity, the angle between the irradiation of the microwave through two intersecting wires placed on the sample, to perform general quantum operations on a degenerate two-level subsystem of spin triplet electrons, called "geometric qubits." .

During nuclear spin manipulation, the researchers set the electron spin state to |+1⟩s and used microwaves with right circular polarization, simplifying the separate manipulation of the nitrogen and carbon nuclear spins. As shown in Figure 1c, the fidelity is increased by reducing the Rabi Frequency; in a specific demonstration, the team used GRAPE (Gradient Rise Pulse Engineering) optimized waveforms to take into account two strongly coupled carbon atoms, The manipulation time is extended to effective T2* with hyperfine coupling below 0.1MHz, resulting in higher fidelity.

Figure 2a shows a quantum circuit that creates entanglement between three nuclear spins (one nitrogen and two carbons). Necessary operational elements for the demonstration are state initialization, universal quantum gates, and state measurement of nuclear spins. The carbon nuclear spin is initialized by the probabilistic projection of the conditional electron spin (so-called measurement-based initialization) with a fidelity of over 99%; on the other hand, due to the presence of depolarization during the photoexcitation of the electron spin measurement , the measurement-based initialization only allows about 90% fidelity, so the nitrogen nuclear spins are deterministic initialization with about 95% fidelity. The Hadamard gate of nuclear spins is achieved by applying π/2 pulses of radio waves resonating with the hyperfine splitting phase of the corresponding carbon nuclear spins, which superimpose the ground states (see Fig. 2b); The correlation operation between the two is achieved by applying a geometric phase-based holographic C-Z gate through the 2π rotation of the Bloch sphere with the help of electron spins (Fig. 2c); finally, in the GHZ state (Greenberger-Horne-Zeilinger state, which has considerable application prospects in the field of quantum information technology because of its maximum entangled properties and measurement accuracy approaching the Heisenberg limit), the joint state of the three nuclear spins is measured in a single time (Fig. 2d), with 78 % fidelity confirms the classical correlation of entanglement.

Figure 2: Three-qubit entanglement generation. a) Quantum circuits for entanglement between three nuclear spins, one nitrogen and two carbon isotopes. This is equivalent to three-qubit quantum error correction in the quantum circuit encoded in Figure 3. H, X, and Y denote Hadamard, Pauli-X, and Pauli-Y gates, respectively. b) The pulse sequence in the experiment. Nuclear spin manipulation is performed by radio frequency (RF), and geometric phase manipulation between electrons (e), nitrogen (N), and carbon (C) is performed by microwave (MW) using the GRAPE algorithm. RO, Init, E_y, and A_1 represent readout, initialization, and two orbital excited states, respectively. c) Conceptual diagram of the "holonomic controlled-phase gate" in encoding. d) GHZ entanglement of three qubits generates a classical evaluation. Correlations were confirmed by measuring the z-axis of all nitrogen and carbon nuclear spins, and since the quantum correlation of three qubits can be measured with the same technique as the quantum correlation of two qubits, the measurement time was not taken into account.

Three-qubit error correction

The three-qubit QEC code implemented in the demonstration is the basic building block of Shor's nine-qubit QEC code and can correct either bit-flip errors (Fig. 3a) or phase-flip errors (Fig. 3b), except for Hadamard, which is inserted immediately after encoding and before decoding doors, they are otherwise identical. In the experiment, errors that occur on the nitrogen nuclear spin are protected by two carbon nuclear spins, and while errors that occur on any single qubit can be corrected after encoding, errors that occur on multiple qubits cannot be corrected.

Figure 3 below shows whether bit-flip and phase-flip errors can be corrected by intentionally inserting errors in the encoded nitrogen nuclear spins. The quantum Toffoli gate of QEC is also configured with a combination of Hadamard gate and holographic C-Z gate. Quantum tomographic measurements of the nitrogen nuclear spin state after QEC show that the state fidelity for bit flip and phase flip errors is on average 75.4% and 74.6%, respectively. %.

Figure 3: Three-qubit quantum error correction (QEC) quantum circuit for bit flip (a) and phase flip (b) errors. The phase-flip code (b) is the same as the bit-flip code (a), except for Hadamard (H) gates inserted immediately after encoding and before decoding. X, Y, and Z denote Pauli X, Pauli Y, and Pauli Z gates, respectively. c, d Bloch spheres representing the nitrogen nuclear spin state through state tomography after quantum error correction (QEC) to prevent bit flip (c) and phase flip (d) errors (left) and nitrogen nuclear spin State fidelity for six initialization states (right). The dashed line represents the estimated fidelity when state preparation and measurement (SPAM) errors are included. Blue (red) bars represent experimental results using QEC. Error bars are defined as the standard deviation of photon shot noise. Since errors do not affect the fidelity in the corresponding base, there is no difference between with or without errors, and these fidelities are even higher without QEC.

Does QEC really work?

To further assess the effectiveness of QEC, the researchers measured the state fidelity of nitrogen nuclear spins when the probability of a single intentional error was changed, and the results are shown in Figure 4. Without QEC, fidelity decreases proportionally to the probability of error. The 10% difference from the ideally obtained fidelity with error probabilities of 0 and 1 should be attributed to state preparation and measurement (SPAM) errors; on the other hand, for QEC, the fidelity is constant regardless of the error probability. Although the fidelity is lower than without QEC due to operational errors when the error probability is below 0.15, it exceeds the fidelity without QEC for error probability above 0.15.

Figure 4: Fidelity of QEC. a) The dependence of nitrogen spin state fidelity is initialized as |+⟩_n on the error probability. The blue (red) solid line is the experimental result of the fidelity obtained using quantum error correction (QEC). The dotted line is the analog fidelity without QEC. Due to operational errors other than state preparation and measurement errors, the fidelity of no QEC is higher at p < 0.15 and the higher fidelity of QEC at p > 0.15. Error bars are defined as the standard deviation of photon shot noise. b) Estimated fidelity degradation for quantum encoding (decoding) operations as a function of hyperfine coupling with another carbon that cannot be detected by optically detected magnetic resonance. In the encoding (decoding) operation, a holographically controlled phase (C-Z) gate with GRAPE-optimized pulses is applied to a certain state, and the operational fidelity is simulated by a trace inner product between the generated and ideal states, the third carbon's presence reduces operational fidelity.

In conclusion, the researchers demonstrate that a three-qubit QEC can prevent bit-flip or phase-flip errors by introducing a holographic C-Z gate of three nuclear spins around a diamond's NV color center at zero magnetic field. The demonstration is suitable for building large-scale distributed quantum computers and long-distance quantum communication networks by connecting quantum systems susceptible to magnetic fields, such as superconducting qubits with spin-based quantum memory.

Link:

[1] https://www.nature.com/articles/s42005-022-00875-6

[2] https://phys.org/news/2022-04-fault-tolerant-quantum-memory-diamond.html