Error protected qubits in a silicon photonic chip

a. Device fabrication

• The circuit is patterned by electron-beam lithography and plasma etching after SiO2 deposition, aluminum-mirror deposition, wafer bonding, and removal of the original substrate and buried oxide.The grating couplers are fully etched, requiring only one silicon etch step.Source: a. Device fabrication · p. 10
• A deposited SiO2 cladding layer is planarized before titanium phase shifters and Au/Ti conducting wires and electrode pads are fabricated.The final top SiO2 cladding is 1 µm thick.Source: a. Device fabrication · p. 10
• The chip-component figure presents waveguides, grating couplers, Mach-Zehnder interferometers, asymmetric interferometers, crossers, and oblong spiral sources.The insets show component images and response or calibration plots; spiral sources are designed to minimise bending losses.Source: Figure 1 · p. 10

b. Device components

• Mach-Zehnder interferometers use 2 × 2 multi-mode interferometers with less than 0.5 dB insertion loss, and titanium micro-heaters provide a 2π phase shift over 0–6 V.The heaters are 1.5 × 100 µm and have an average resistance of 500 Ω.Source: b. Device components · p. 11
• The asymmetric Mach-Zehnder interferometers route signal and idler frequencies to separate output ports, with a typical filter extinction ratio of around −40 dB.Their arm-length difference is 22.7 µm for the selected frequencies, separated by approximately 10 nm.Source: b. Device components · p. 11
• Crosser arrays route signal and idler photons among relative qudit modes, with measured loss of −0.1 dB and crosstalk of -40 dB.Source: b. Device components · p. 11
• The optical circuit measures approximately 0.2 cm × 0.8 cm and contains 48 thermo-optic titanium phase modulators and more than 200 passive integrated components.Average single-photon channel efficiency from source to detector was approximately −15 dB (3%).Source: b. Device components · p. 11
• The chip schematic combines eight photon-pair sources, integrated filters that spatially separate signal and idler photons, entangling gates, and Mach-Zehnder interferometers for qudit measurements.The resulting resource is a four-photon, four-dimensional entangled state.Source: Figure 2 · p. 11

c. Experimental set-up

• Pump pulses at 1541.35 nm and an average launched power of 11.2 mW generate photons collected through filtered fiber channels and detected by four superconducting nanowire detectors.The sources’ typical pair-generation probability is 0.03 and standard four-fold coincidence rates are approximately 1 Hz.Source: c. Experimental set-up · p. 12

a. Phase shifter characterisation

• Each phase shifter is calibrated by measuring its current-voltage relationship and fitting a bright-light interferometer fringe to infer the phase-voltage relation.The calibration fringes average 99% visibility; thermal cross-talk was negligible, and Peltier-based PID control prevents phase drifts due to bulk heating.Source: a. Phase shifter characterisation · pp. 12–13

b. Source characterisation

• RHOM interference fringes between source pairs all had visibilities above 0.88, indicating a good degree of source indistinguishability.Narrow off-chip filters are used to address spectral indistinguishability.Source: b. Source characterisation · p. 14
• For sources S1 and S2 at ∼10 mW input power and ∼3% pair-generation probability, HHOM fringe visibility was 0.86 ± 0.04, corresponding to HOM-equivalent visibility 0.84 ± 0.05.The fringe depth relates to a traditional HOM dip and thus total photon indistinguishability, lower-bounding purity.Source: b. Source characterisation · p. 14

3. MEASUREMENTS PERFORMED FOR 4P4D STATE FIDELITY

• The 4P4D-state fidelity calculation requires diagonal computational-basis terms and off-diagonal density-matrix elements, which are measured through symmetric and antisymmetric operators.The off-diagonal terms are evaluated using eigenbases formed from paired basis states and their ±i combinations.Source: §3 · pp. 14–15
• Stabiliser measurements reduce the 4P4D fidelity protocol to 5,600 projectors for the off-diagonal terms, with 256 computational-basis projectors used for the diagonal overlap.The protocol uses GHZ-state stabilisers in effective two-mode subspaces for the relevant mode pairs.Source: §3 · p. 15
• The fidelity procedure estimates only 312 of the 65,536 density-matrix elements and is not a full state-reconstruction method.Source: §3 · p. 15

• The supplementary table lists the operators measured for each four-qubit star stabilizer.Source: Table 1 · p. 16
• Four-photon graph states are derived from the eight-qubit star using intra-qudit controlled-phase gates, local complementation, and Z measurements; the clover hypergraph state is also generated and verified.Source: §4 · p. 16
• Arbitrary qudit local operations and projective measurements provide a toolkit for generating a wide variety of graph states.Source: §4 · p. 16
• Local qubit operations enable transformations within entanglement classes, same-photon qubit-pair CZ gates toggle edges, and Z measurements remove vertices.Source: §4 · p. 16
• Local complementation transforms graph edges by toggling edges among a chosen vertex’s neighbors.Source: §4 · p. 16

• Two-photon state-generation data collection is orders of magnitude faster than four-photon experiments, and the experiments verify local qudit unitaries encoding four qubits in two photons.Source: a. Two-photon, four-qubit star · p. 16
• Pumping sources S1 and S4 and post-selecting one photon per qudit generates a two-photon, four-qubit star graph state.Source: a. Two-photon, four-qubit star · p. 16
• Applying H2H3H4 yields |star4⟩, whose experimental state was reconstructed using compressed sensing.The authors attribute the feasibility of reconstruction to the high data-collection rate using two photons.Source: a. Two-photon, four-qubit star · p. 17

b. Two-photon, four-qubit linear cluster

• 0.82 ± 0.02 was the fidelity of the two-photon four-qubit line graph state against the ideal state.The authors attribute its lower fidelity than the star state to the four-source pumping scheme and its source overlap and relative calibration.Source: Figure 5 · p. 17, b. Two-photon, four-qubit linear cluster · p. 18

c. Eight-qubit star

• The eight-qubit star is obtained from an eight-qubit GHZ state by Hadamards, and Z measurements can remove vertices to produce star graphs with n ≤ 8 qubits.The state is verified using θ-measurements of population and coherence.Source: c. Eight-qubit star · p. 18

d. Linear cluster states

• Local CZ operations, local complementations, and qubit projections derive five-, four-, and three-qubit linear clusters from the eight-qubit star.The four-qubit box cluster is locally equivalent to the four-qubit line.Source: d. Linear cluster states · p. 19

e. Double-branched and crazy graph states

• The three-, five-, and seven-qubit branched states are derived from the eight-qubit star; the six-qubit crazy graph is generated in the same way.The supplied text describes stabiliser measurements for verification but gives no measured fidelity values for these states.Source: e. Double-branched and crazy graph… · pp. 19–20, Figure 8 · p. 21, Table 5 · p. 22

f. Accessing graph state entanglement classes

• A random search over local complementations, qubit removals, and selected CZ operations found 21 stabiliser entanglement types and 127 non-isomorphic graph states accessible on the device.The search began with the eight-qubit star graph and continued until no new entanglement classes were found.Source: f. Accessing graph state entanglement… · p. 23, Figure 9 · p. 23

g. Hypergraphs

• The clover hypergraph is derived by projecting qubits 2, 4, and 8 from an eight-qubit graph state, while projections also produce Toffoli and fully connected Toffoli hypergraphs.The clover state starts from the four-photon, four-dimensional entangled state after pumping sources S1 through S8 and turning on every intra-qubit gate.Source: g. Hypergraphs · p. 24

5. MEASUREMENT-BASED PROCESS TOMOGRAPHY: ENCODING ARBITRARY INITIAL STATES

• For arbitrary initial states in a linear-cluster MBQC process, the paper shows an encoding measurement procedure equivalent to initializing the leftmost qubit in the desired state.The measurement choice must account for whether the initial-state basis is conjugated.Source: §5 · pp. 25–26

a. Euler decomposition for measurement-based one-qubit rotations

• Process tomography on the three-qubit line is limited to RX rotations, while the crazy-graph encoding discards cases with a single physical phase error rather than correcting them.Encoding the logical |0⟩ state requires an entangled-basis measurement of physical qubits within the same qudit.Source: a. Euler decomposition for… · p. 26, b. Logical Pauli bases: the crazy… · p. 27

c. Process tomography results

• Supplementary Table 12 summarizes measured process fidelities for the physical three-qubit line and logical six-qubit crazy graph implementations.The supplied table caption does not include the fidelity values.Source: Table 12 · p. 27, c. Process tomography results · p. 27

7. MEASUREMENT-BASED PHASE ESTIMATION.

• Phase estimation maps a circuit onto measurements of a three-qubit line, with each node measurement implementing an H RZ(θ) gate.The leftmost and central measurement angles determine the sequential operations, and the rightmost outcome infers a phase bit.Source: §7 · pp. 27–28
• The initial phase-estimation measurement uses φ = 0 to initialize the computation in |+⟩, and subsequent angles are chosen iteratively from measurement outcomes.The resulting phase reconstruction process is described as identical to the circuit representation.Source: §7 · p. 28

8. MAPPING PHYSICAL ERRORS TO COMPUTATIONAL ERRORS

• The explored error-correcting codes detect only a subset of possible device-noise channels, and the simulations focus on source spectral distinguishability and imperfect phase-shifter control.The authors note that these physical errors can include coherent and incoherent contributions.Source: §8 · p. 28
• Spectral-distinguishability computational errors are independent of whether physical or logical graph encodings are used.The noise study models phase-shifter imperfections and spectral distinguishability separately.Source: §8 · p. 28
• Approximately 4% enhancement was estimated for logical graph encoding under typical phase-voltage noise values.The authors model each phase shifter's voltage as an independent Gaussian random variable centered on its ideal voltage and repeat simulations 500 times.Source: §8 · p. 28

9. LOSS TOLERANCE IN CRAZY GRAPH STRUCTURES

• The loss challenge worsens as photon number grows because larger entangled states require larger circuits and more photons, while post-selected generation probability decreases exponentially.The passage identifies photon loss and post-selection as scaling concerns.Source: §9 · p. 28
• Post-selection filters photon loss but requires all four photons to be detected in their qudit modes, reducing the four-fold coincidence rate.The graph state does not exist before all photons are detected in this scheme.Source: §9 · p. 28
• Simulations compare physical and logical encodings under phase-voltage noise and distinguishability noise, with no encoding difference reported for the distinguishability model.The phase-voltage comparison includes 95% confidence intervals from 500 independent noisy-voltage simulation sets per data point.Source: Figure 11 · p. 29
• Heralded crazy graphs can retain an entanglement path despite up to n −1 lost physical qubits in each n-qubit vertical column, while heralded integrated-photonics approaches remain under development.Such schemes can leave graph states in an event-ready form before measurement.Source: §9 · p. 29
• Teleportation fidelity for |−i⟩ did not substantially change when information from one intermediate photon was discarded, even with phase errors applied to intermediate physical qubits.The experiment simulated loss by discarding a photon’s measured information after all four photons had been detected for post-selection.Source: §9 · p. 29

10. DATA ANALYSIS FOR COMPARING PEA LOGICAL AND PHYSICAL IMPLEMENTATIONS

• The Monte Carlo analysis resamples each bit’s measured zero and one counts with binomial statistics, recalculates phases, and compares correct-phase counts for logical and physical data.The procedure repeats for N = 10^4 rounds and estimates confidence from the fraction of rounds won by the logical case.Source: §10 · pp. 29–30
• A separate branched-state simulation reports successful teleportation despite multiple qubits lost from one photon, including cases with phase-error probabilities p = 0.2 and p = 0.4.The probabilities are applied to intermediate physical qubits.Source: Figure 12 · p. 30
• 98% is the estimated confidence level that the logical implementation provides higher phase-estimation success rates than the physical implementation.The estimate is based on the data in Fig. 4c of the main text.Source: §10 · p. 30

11. QUDIT ENCODINGS SCALING FOR CURRENT QUANTUM PHOTONIC IMPLEMENTATIONS

• Post-selection exponentially reduces coincidence rates as photon number increases, but it is used here to test the platform rather than as an essential architectural requirement.The passage notes heralding and multiplexing as alternatives under development.Source: §11 · p. 30
• The rate estimates account for probabilistic fusion gates, probabilistic photon-pair generation, and photonic-circuit losses when comparing equal-qubit-count states.The described post-selected fusion success probability is p = 1/2, and local operations on d dimensions require d −1 MZIs with transmission decreasing as η^(d−1).Source: §11 · pp. 30–31
• Qudit encoding provides a feasible route to states of up to 40 qubits without processing more than 10 photons, using dimensions up to d = 16.The rate estimates report significant advantages for high-dimensional encoding at a given technology and photon count; Supplementary Figure 13 compares n-qubit GHZ rates across schemes and sizes.Source: §11 · p. 31, Figure 13 · p. 31
• The estimates use both current-device components and a low-loss-component scenario, with the same experimental pumping scheme of s ∼0.03 and a 500 MHz repetition rate.The low-loss scenario includes directional couplers with 1% loss and high-heralding-efficiency sources.Source: §11 · p. 31