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

• is projected on with probability

• We know to which subspace was projected

von Neumann Measurementand orthogonal subspaces,

is the orthogonal projection on is the orthogonal projection on

1 2 k… k+1 n…

information qubitsin state

n1 2 …

quantum codewordin the state

unitary rotation

Quantum Codes

redundant qubitsin 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 ofunder 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 ofunder 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 aunitary 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

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

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: