Fidelity of Finite Length Quantum Codes in Qubit Erasure Channel - - PowerPoint PPT Presentation

Fidelity of Finite Length Quantum Codes in Qubit Erasure Channel Alexei Ashikhmin, Bell Labs

▪ Correction of Erasures by Classical Codes ▪ Quantum Erasure Channel (QEC) ▪ Probability of Decoding Error in QEC via Combinatorial Invariants ▪ MacWilliams Type Identities for the Combinatorial Invariants ▪ Linear Programming Bound on Probability of Decoding Error ▪ Linear Programming Bound on the Number of Unrecoverable Sets of i Erasures

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Correction of Erasures by Classical Codes (1)

Binary Erasure Channel 1 1 (erasure)

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Correction of Erasures by Classical Codes (2)

• Let

be a generating matrix of an [n,k] linear code C

• We transmit
• Let erasures occur on positions so we get

a vector of the form When we can recover erasures on

• Let be the set of non-erasures
• We can recover erasures on iff columns of G corresponding to

non erased positions have full rank, i.e.,

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• In the best possible case we have that any k columns of G have rank k
• In this case only if erasures are not recoverable
• Thus we have the following simple, but tight, lower bound:

Theorem 1 (known of course):

• This simple bound leads to Gallager’s spherical packing bound on the

error exponent:

Combinatorial Lower Bound on Probability of Decoding Error

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Quantum Erasure Channel (QEC) (1)

• Erasure means that qubit is moved into state which is orthogonal

to legitimate states and . Hence, can be detected: encoding

Quantum Erasure Channel

n code qubits k inf. qubits

• Thus, we can identify the set of positions S of qubits erasures
• Erasures occur with probability

erased qubits

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2k qubits in maximally entangled state

Quantum Erasure Channel (QEC) (2)

Encoder Decoder QEC QEC 1 n 1 n 2k qubits in a new state , which depends on S A standard way to characterize the performance of a quantum code Q: entangle k information qubits with k qubits of a reference system Uhlmann's fidelity:

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Quantum Erasure Channel (QEC) (3)

• A set of erasures S is recoverable if Uhlmann's fidelity is 1, i.e.,

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• We define the probability of decoding error by

is symplectic inner product and is defined by:

• An [[ n, k ]] stabilizer code Q is a subspace of dimension in
• Q is associated with a classical [n, (n+k)/2] code over
• The code has the property that its dual code , i.e.

Quantum Linear Stabilizer Codes

Here is defined by , where

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Here denotes a vector from An alternative way of defining recoverable erasures is more useful:

• 1. Code is a union of cosets of :
• 2. We transmit a codeword and get (symbols from S are erased)

Theorem 2: S is recoverable iff belongs to exactly one coset

• Note that it is not necessary to restore the codeword itself; we need

to identify only the coset

Probability of Decoding Error (1)

code (coset ) coset coset code

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• For a vector and a set of positions S, the punctured vector is

Example: If then

• For a code we define punctured and shortened codes:

and Theorem 3: belongs to exactly one code coset (and hence S is recoverable) iff

• Motivated by the above observation we define the code invariants:

Combinatorial Characterization of (2)

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Theorem 4: The number of recoverable sets of erasures is equal to Equivalently, the number of unrecoverable sets is (of course ) Corollary 5:

Combinatorial Characterization of (3)

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Example 1. Let Q’ be the [[12, 6]] code from the table of the best known

• codes. Q’ can be defined by the parity check matrix of code
• Using this matrix we can find that
• Using these we find the number of unrecoverable sets of size i :
• Note that is visibly smaller than which is the number of

all 3-erasure sets

Combinatorial Characterization of (4)

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Example 1 (continued) Using we can compute for any

Combinatorial Characterization of (5)

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• We define vector and matrix with

where and defined recursively (see the paper) Theorem 6: Note that in

MacWilliams Type Identities

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• First, we establish the relation between and

Theorem 7: (1) where (2)

• We define vector with
• (1) and (2) show that any is a linear combination of -s and

therefore (1) and (2) allow us to form the matrix so that

• Since all we get the constraint:

Lower Bounds on and (1)

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• Now we can formulate the following linear programming problem

(3)

• Let me remind that
• This shows that is a linear combination of -s and, therefore, we

can form vector so that

• Using this in (3) and solving the linear programming problem, we get a

lower bound

Lower Bounds on and (2)

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• Similarly, we can define so that

(4) where is the number of uncorrectable sets of i erasures, and obtain a lower bound on Example 2 Let n=12 and k=6. 1. Solving the optimization problem (3) with defined in (4), we prove that any [[12,6]] code (nondegenerate or degenerate) has Let me remind that the [[12,6]] code Q’ (with the largest min. distance) has

Lower Bounds on and (3)

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Example 2 (continued) 2. Solving the optimization problem (3) with defined so that we find such that any [[12,6]] code (nondegenerate or degenerate) has

• 3. Previously known lower bound:
• M. Tomamichel, M. Berta, J. M. Renes, “Quantum Coding with Finite

Resources,” Nature Communications, 7, 2016: 11419 the following lower bound was obtained (20) This bound is quite weak, as the figure on the next slide shows

Lower Bounds on and (4)

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The exact for [[12,6]] code Q’, our new bound, and previously known bound (20)

Lower Bounds on and (5)

Example 2 (continued)

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Thank You for Your Attention

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