Blind topological measurement-based quantum computation

Secure delegated computation is already in the practical phase for classical computing,

• Blind quantum computation lets Alice delegate a computation to Bob while preventing him from learning her input, output, and algorithm.The paper notes that this protocol was experimentally demonstrated in an optical system.Source: §2 · p. 3
• The protocol has an error threshold of 4.3 × 10−3, close to the 7.5 × 10−3 threshold of non-blind topological quantum computation.The authors also report fault tolerance against detectable qubit loss, including photon loss and escape from the qubit energy level.Source: Secure delegated computation is… · p. 3

The topologically-protected measurement-based quantum computation (TMBQC) [12,

• The RHG lattice is a three-dimensional cubic lattice whose graph state is the resource for topologically protected measurement-based quantum computation.The paper describes X, Z, T, and Y measurements as sufficient for universal TMBQC.Source: The topologically-protected… · pp. 3–4

B. Blind TMBQC

• Adding two qubits to each RHG-lattice qubit creates a decorated lattice that lets Bob simulate X, Y, T, and Z measurements without knowing which basis he is enacting.The protocol uses single-qubit measurements in {|0⟩ ± e^iφ|1⟩} bases.Source: B. Blind TMBQC · p. 4

quantum channel, where

• In the decorated RHG lattice protocol, Alice remembers the random numbers associated with the N qubits used and keeps those numbers secret from Bob, according to the cited description.Source: quantum channel, where · p. 5

Step 2. Now Bob has {|θj⟩}N

• Bob applies CZ gates on the decorated RHG lattice's red edges to create |CΘ⟩, a rotated graph state.The protocol describes this as its second step.Source: Step 2. Now Bob has {|θj⟩}N · p. 5

calculates

• For each qubit, Alice computes a measurement angle using prior outcomes and a random bit, sends it to Bob, and processes returned results for error correction.Bob measures in the instructed equatorial basis and returns each outcome through the classical channel.Source: Step 2. Now Bob has {|θj⟩}N · p. 5, calculates · p. 5

C. Correctness

• Measurements on three decorated-lattice qubits with angles (0, 0, 0) are equivalent to an X-basis measurement on the corresponding original-lattice qubit.The pre-rotation cancels θ_j, while r_jπ only flips the measurement result.Source: C. Correctness · p. 6
• Angle triples (0, 0, π/2), (0, 0, π/4), and (π/2, π/2, π/2) simulate Y-, T-, and Z-basis measurements, respectively.Together with the X-basis sequence, these simulate the original TMBQC using only {|0⟩ ± e^iφ|1⟩} measurements.Source: C. Correctness · p. 6

probability distribution of Alice’s computational angles, given all the classical information

• Given the classical information and outcomes of any POVMs Bob performs at any protocol stage, Alice’s computational-angle distribution remains its prior distribution.The result addresses Bob's possible measurements on his system during the protocol.Source: probability distribution of Alice’s… · p. 6

errors occur during the preparation of initial states {|θj⟩}N

• The protocol models initial-state preparation errors as partial depolarization and includes noisy measurements and CZ gates in its error analysis.The preparation error probability is pP; the measurement and CZ noise models are also described as partially depolarizing.Source: errors occur during the preparation of… · p. 7
• Rotating the local reference frames absorbs the preparation-state rotations, so the threshold calculation can assume all θj = 0.The paper states this simplification holds without loss of generality.Source: errors occur during the preparation of… · p. 7

E. Threshold

• The resource state is assumed to be created stepwise, with an additional fifth step in the blind protocol.The analysis first calculates the induced single-qubit Z errors on the three-qubit chain before local measurements begin.Source: errors occur during the preparation of… · p. 7

• For the (0, 0, 0) measurement pattern, the accumulated error includes λ1, λ3, and two measurement-error contributions of 2pM/3 each.The analysis considers this pattern because it corresponds to an effective X measurement on the original RHG qubit.Source: §3 · p. 7

two-qubit error probability [18] q2 = 4p2/15 + O(p2

• 4.3 × 10−3 is the bulk topological error threshold when pP = pM = p2 = p.The threshold curve for (q1, q2) was numerically calculated using minimum-weight-perfect-matching decoding.Source: two-qubit error probability [18] q2 =… · p. 8
• Defect creation does not lower the blind setup’s threshold because its added Z-measurement error remains below the defect-surface error considered in the referenced analysis.At pM = pP = p2 = 4.3 × 10−3, the added error probability is λZ = 1.6 × 10−2; the threshold is therefore determined by the bulk region.Source: two-qubit error probability [18] q2 =… · p. 8

where ǫY

• With pM = pP = p2 = 4.3 × 10−3, the injected magic-state error is 0.043, below the distillation thresholds ǫY = 0.38 and ǫT = 0.17.The paper attributes the injected-state error to CZ gates, measurements, and state preparations in the decorated resource.Source: where ǫY · p. 9
• 1.6×10−3 is the threshold when pP = 10p and pM = p2 = p, showing robustness to larger preparation errors.The paper notes that preparation errors behave as independent errors, which are known to be easier to correct.Source: where ǫY · p. 9

• For independent qubit loss, the blind protocol’s threshold follows the referenced TMBQC result by replacing ploss with 3ploss.The three-qubit chain is treated as lost if any one of its qubits is lost; a post-selected scheme has overhead ≃(1 −3ploss)d3, independent of algorithm size.Source: §II · p. 10
• A similar fusion strategy can create the decorated resource state from puffer-ball states, allowing a threshold analysis for nondeterministic CZ gates.The cited example adds a one-dimensional chain of two qubits to each puffer-ball root qubit.Source: §II · p. 10
• Earlier brickwork-state work gave only an existence proof, while the paper states its threshold should be lower than that of TMBQC.The discussion gives estimated thresholds of order 10−5 for the circuit-model scheme and ∼10−6 when implemented on a two-dimensional brickwork state; an explicit threshold for the cited three-dimensional scheme was unknown.Source: §II · p. 10
• Minimum-weight-perfect-matching decoding scales polynomially but may require substantial classical resources; other algorithms can decode four million qubits within seconds.The cited algorithm’s threshold is 0.9%, compared with 0.75% for the referenced result.Source: §II · p. 10

A. Definition of the blindness

• Blindness requires Bob’s conditional distribution over Alice’s computational angles to remain their prior distribution, and the final classical output to be one-time padded.The condition covers classical information and outcomes of any POVM Bob may perform on his system during the protocol.Source: A. Definition of the blindness · p. 11

B. Our protocol satisfies B1

• The B1 analysis defines the protocol’s sent angles, secret computational angles, random prerotations, and hidden binary parameters, related by Δj = Φj + Θj + Rjπ (mod 2π).These variables specify how Alice’s secret angles and randomization determine the angles sent to Bob.Source: B. Our protocol satisfies B1 · p. 11

POVM which Bob may perform on his {|θj⟩}N

• Bob’s knowledge of Alice’s angles is represented by their conditional distribution given his POVM outcome and the angles sent by Alice.The passage introduces Bayes’ theorem as the next step in the analysis.Source: POVM which Bob may perform on his… · pp. 11–12

C. Our protocol satisfies B2

• The state after measuring the third qubit is one-time padded with Z^s1X^s2, where s1 and s2 are the first and second qubits’ measurement results.The section states that these results are unknown to Bob because the random bits rj are unknown to him.Source: C. Our protocol satisfies B2 · p. 12

are unknown to Bob. We can show that {rj}N

• The cited fragment states that j=1 are unknown to Bob, but it does not identify the referent or establish a one-time-pad argument.Source: are unknown to Bob. We can show that… · p. 12

Figure 1

• Figure 1 contrasts the elementary cell of the RHG lattice L with that of the decorated lattice L′.The latter is the lattice used for the decorated construction.Source: Figure 1 · p. 15

Figure 2

• Figure 2 is captioned as illustrating how the decorated lattice works, while the supplied passage associates three-qubit measurements with X, Y, T, and Z basis measurements.The measurement angles listed correspond to those four simulated bases.Source: Author information Correspondence and… · p. 16, Figure 2 · p. 16

Figure 3

• Figure 3 depicts the topological blind protocol, including Alice sending Bob randomly rotated qubits and the protocol’s measurement-and-message exchange.Bob creates the decorated lattice, Alice sends a classical message, and Bob measures and returns the result.Source: Figure 3 · p. 17, Author information Correspondence and… · p. 17

Figure 4

• Figure 4 shows resource-state creation in steps (a)–(e), and the threshold analysis includes an additional fifth step.The passage identifies the stepwise construction as the assumed resource-state preparation.Source: Figure 4 · p. 17

Figure 5

• Figure 5 presents a measurement pattern in which three qubits are measured in numerical order.The corresponding three-qubit sequence is described as running from bottom to top.Source: Figure 5 · p. 17