Fidelities of Quantum ARQ Protocol Alexei Ashikhmin Bell Labs  Classical Automatic Repeat Request (ARQ) Protocol  Qubits, von Neumann Measurement, Quantum Codes  Quantum Automatic Repeat Request (ARQ) Protocol  Quantum Errors  Quantum Enumerators  Fidelity of Quantum ARQ Protocol Quantum Codes of Finite Lengths The asymptotical Case (the code length ) Some results from the paper “Quantum Error Detection”, by A. Ashikhmin,A. Barg, E. Knill, and S. Litsyn are used in this talk

is a parity check matrix of a code Compute syndrome If we detect an error If, but we have an undetected error Classical ARQ Protocol Noisy Channel

The state (pure) of qubits is a vector Manipulating by qubits, we effectively manipulate by complex coefficients of As a result we obtain a significant (sometimes exponential) speed up qubits Qubits

In this talk all complex vectors are assumed to be normalized, i.e. All normalization factors are omitted to make notation short

is projected on with probability We know to which subspace was projected von Neumann Measurement and orthogonal subspaces, is the orthogonal projection on

12k … k+1n … information qubits in state n 12 … quantum codeword in the state unitary rotation Quantum Codes redundant qubits in the ground states is the code space is the code rate the joint state:

ARQ protocol: –We transmit a code state –Receive –Measure with respect to and –If the result of the measurement belongs to we ask to repeat transmission –Otherwise we use Quantum ARQ Protocol is fidelity If is close to 1 we can use

Conditional Fidelity The conditional fidelity is the average value of under the condition that is projected on Recall that the probability that is projected on is equal to Quantum ARQ Protocol

Quantum computer is unavoidably vulnerable to errors Any quantum system is not completely isolated from the environment Uncertainty principle – we can not simultaneously reduce: –laser intensity and phase fluctuations –magnetic and electric fields fluctuations –momentum and position of an ion The probability of spontaneous emission is always greater than 0 Leakage error – electron moves to a third level of energy Quantum Errors

Depolarizing Channel (Standard Error Model) Depolarizing Channel means the absence of error are the flip, phase, and flip-phase errors respectively This is an analog of the classical quaternary symmetric channel Quantum Errors

Similar to the classical case we can define the weight of error: Obviously Quantum Errors

Quantum Enumerators P. Shor and R. Laflamme: is a code with the orthogonal projector

and are connected by quaternary MacWilliams identities where are quaternary Krawtchouk polynomials: The dimension of is is the smallest integer s. t. then can correct any errors Quantum Enumerators

In many cases are known or can be accurately estimated (especially for quantum stabilizer codes) For example, the Steane code (encodes 1 qubit into 7 qubits): Quantum Enumerators and therefore this code can correct any single ( since ) error

Fidelity of Quantum ARQ Protocol Theorem The conditional fidelity is the average value of under the condition that is projected on Recall that the probability that is projected on is equal to

Lemma (representation theory) Let be a compact group, is a unitary representation of, and is the Haar measure. Then Lemma

Quantum Codes of Finite Lengths We can numerically compute upper and lower bounds on, (recall that ) Fidelity of the Quantum ARQ Protocol

Sketch: using the MacWilliams identities we obtain using inequalities we can formulate LP problems for enumerator and denominator

For the famous Steane code (encodes 1 qubit into 7 qubits) we have: Fidelity of the Quantum ARQ Protocol

Lemma The probability that will be projected onto equals Hence we can consider as a function of Fidelity of the Quantum ARQ Protocol

Let be the known optimal code encoding 1 qubit into 5 qubits Let be code that encodes 1 qubit into 5 qubits defined by the generator matrix: is not optimal at all Fidelity of the Quantum ARQ Protocol

Theorem ( threshold behavior ) Asymptotically, as, we have Theorem (the error exponent) For we have The Asymptotic Case Fidelity of the Quantum ARQ Protocol (if Q encodes qubits into qubits its rate is )

Existence bound Fidelity of the Quantum ARQ Protocol Theorem There exists a quantum code Q with the binomial weight enumerators: Substitution of these into gives the existence bound on Upper bound is much more difficult

Fidelity of the Quantum ARQ Protocol Sketch: Primal LP problem: subject to constrains:

Fidelity of the Quantum ARQ Protocol From the dual LP problem we obtain: Theorem Let and be s.t. then Good solution: