Hardness of correcting errors on a Stabilizer code (arXiv:1310.3235) - - PowerPoint PPT Presentation

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Hardness of correcting errors on a Stabilizer code

(arXiv:1310.3235) Pavithran Iyer, Maˆ ıtrise En Physique, Superviseur: David Poulin, Universit´ e de Sherbrooke Fall INTRIQ meeting, November 5th−6th, 2013

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

In this talk . . .

1 Computational Complexity 2 Classical error correction 3 Quantum error correction 4 Main result 5 Conclusions

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Contents of this talk

1 Computational Complexity 2 Classical error correction 3 Quantum error correction 4 Main result 5 Conclusions

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Easy and hard problems:

Some problems are easy → we can solve them “efficiently”: Ex. Arithmetic operations, . . . P: All problems that can be solved in polynomial-time (polynomial in input size)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Easy and hard problems:

Some problems are easy → we can solve them “efficiently”: Ex. Arithmetic operations, . . . P: All problems that can be solved in polynomial-time (polynomial in input size) Often, we do not have an efficient solution. But we can verify any proposal in poly-time.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Given all the fares for travel . . . Is there any way of touring India that costs ≤ $2000 ? Too many options for a brute-force search !

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Given all the fares for travel . . . Is there any way of touring India that costs ≤ $2000 ? Too many options for a brute-force search ! Given a proposal for a tour, it is easy to verify if it costs ≤ $2000.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Given all the fares for travel . . . Is there any way of touring India that costs ≤ $2000 ? Too many options for a brute-force search ! Given a proposal for a tour, it is easy to verify if it costs ≤ $2000. NP: All problems such that any certificate can be verified in polynomial-time.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Given all the fares for travel . . . Is there any way of touring India that costs ≤ $2000 ? Too many options for a brute-force search ! Given a proposal for a tour, it is easy to verify if it costs ≤ $2000. NP: All problems such that any certificate can be verified in polynomial-time. Some problems need a lot of effort → if we can solve them, we can solve any NP problem.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Given all the fares for travel . . . Is there any way of touring India that costs ≤ $2000 ? Too many options for a brute-force search ! Given a proposal for a tour, it is easy to verify if it costs ≤ $2000. NP: All problems such that any certificate can be verified in polynomial-time. Some problems need a lot of effort → if we can solve them, we can solve any NP problem. NP-Complete: Problems whose solution can be used to solve any NP problem in poly-time.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Sometimes we are not happy with just one solution . . . want to know how many are there ? How many tours are there of India that cost ≤ $2000 ? Brute-force search is Hopelessly hard !

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Sometimes we are not happy with just one solution . . . want to know how many are there ? How many tours are there of India that cost ≤ $2000 ? Brute-force search is Hopelessly hard ! We are now counting the number of solutions to previous NP problem.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Sometimes we are not happy with just one solution . . . want to know how many are there ? How many tours are there of India that cost ≤ $2000 ? Brute-force search is Hopelessly hard ! We are now counting the number of solutions to previous NP problem. #P: All problems that involve counting solutions to a NP problem.

#P-Complete: Problems whose solution can be used to solve any #P problem in polynomial-time.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Contents of this talk

1 Computational Complexity 2 Classical error correction 3 Quantum error correction 4 Main result 5 Conclusions

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Hard problems in classical error correction

Classical information is encoded and transmitted in bits → strings of 0’s and 1’s. Consider a simple code: C = { A 000, B 111}. If r = 001 is received → some bit(s) were

• flipped. which ones ? ↔ what was added ?

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Hard problems in classical error correction

Classical information is encoded and transmitted in bits → strings of 0’s and 1’s. Consider a simple code: C = { A 000, B 111}. If r = 001 is received → some bit(s) were

• flipped. which ones ? ↔ what was added ?
• e = 001 ↔ Last bit flipped: Pr(

e) ∼ p

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Hard problems in classical error correction

Classical information is encoded and transmitted in bits → strings of 0’s and 1’s. Consider a simple code: C = { A 000, B 111}. If r = 001 is received → some bit(s) were

• flipped. which ones ? ↔ what was added ?
• e = 001 ↔ Last bit flipped: Pr(

e) ∼ p

• e = 110 ↔ first two bits flipped: Pr(

e) ∼ p2

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

A short hand notation . . .

Take the same code: C = {000, 111}.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

A short hand notation . . .

Take the same code: C = {000, 111}. Don’t store the code → properties of the strings “Checks” → point to bits which sum to zero.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

A short hand notation . . .

Take the same code: C = {000, 111}. Don’t store the code → properties of the strings “Checks” → point to bits which sum to zero.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

A short hand notation . . .

Take the same code: C = {000, 111}. Don’t store the code → properties of the strings “Checks” → point to bits which sum to zero.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

A short hand notation . . .

Take the same code: C = {000, 111}. Don’t store the code → properties of the strings “Checks” → point to bits which sum to zero. Syndrome (s) → 0 (satisfied), 1 (not satisfied). r fails to satisfy: r = c + e, where c ∈ C and e / ∈ C.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

What is the binary sequence that satisfies first check and violates the second ?

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

What is the binary sequence that satisfies first check and violates the second ? Choices: e = 001 (last bit flipped)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

What is the binary sequence that satisfies first check and violates the second ? Choices: e = 001 (last bit flipped) OR e = 110 ↔ First two bits flipped.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

What is the binary sequence that satisfies first check and violates the second ? Choices: e = 001 (last bit flipped) OR e = 110 ↔ First two bits flipped.

• e: minimum bit-flips error giving the syndrome.

∴ e = 100.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Another example . . .

Consider a slightly complicated code:

• r is received with s = 010. What is e ?

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Another example . . .

Consider a slightly complicated code:

• r is received with s = 010. What is e ?
• e = 10000 ↔ first bit was flipped

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Another example . . .

Consider a slightly complicated code:

• r is received with s = 010. What is e ?
• e = 10000 ↔ first bit was flipped
• e = 00011 → two bits flipped

(Wrong)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Another example . . .

Consider a slightly complicated code:

• r is received with s = 010. What is e ?
• e = 10000 ↔ first bit was flipped
• e = 00011 → two bits flipped

(Wrong) Given a syndrome → Always look for the sequence corresponding to least bit flips

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

A real world example . . .

Consider a real-life code. Given the syndrome s, what is the error e ? (min bit flips for s)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

A real world example . . .

Consider a real-life code. Given the syndrome s, what is the error e ? (min bit flips for s) Too many (exponential) errors with the same syndrome s → a naive optimisation is hard

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

A real world example . . .

Consider a real-life code. Given the syndrome s, what is the error e ? (min bit flips for s) Too many (exponential) errors with the same syndrome s → a naive optimisation is hard What are the problems of interest ?

1 Given a graph G and

s, determine e of lowest weight for s. (NP-Complete)

2 Given a graph G,

s and i, determine how many e of weight i for s. (#P-Complete)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Contents of this talk

1 Computational Complexity 2 Classical error correction 3 Quantum error correction 4 Main result 5 Conclusions

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

Quantum information is encoded and transmitted in qubit states: |000001, |010101, . . . Errors: independent bit flips X, phase flips Z on each qubit. (Independent X − Z channel)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

Quantum information is encoded and transmitted in qubit states: |000001, |010101, . . . Errors: independent bit flips X, phase flips Z on each qubit. (Independent X − Z channel) Extend the example of the classical code “Checks” are properties we can verify without disturbing the state (measure)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

Quantum information is encoded and transmitted in qubit states: |000001, |010101, . . . Errors: independent bit flips X, phase flips Z on each qubit. (Independent X − Z channel) Extend the example of the classical code “Checks” are properties we can verify without disturbing the state (measure) S =< IXY II, ZIIY I, IIY Y Y >. s: 0 (commutes) and 1 (anti-commutes)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

If s = 010, what is E ? (that affects the least number of qubits)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

If s = 010, what is E ? (that affects the least number of qubits) E = XIIII ↔ Only 1st qubit flipped.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

If s = 010, what is E ? (that affects the least number of qubits) E = XIIII ↔ Only 1st qubit flipped. XXY II = S1 · XIIII,

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

If s = 010, what is E ? (that affects the least number of qubits) E = XIIII ↔ Only 1st qubit flipped. XXY II = S1 · XIIII, Y IIY I = S2 · XIIII,

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

If s = 010, what is E ? (that affects the least number of qubits) E = XIIII ↔ Only 1st qubit flipped. XXY II = S1 · XIIII, Y IIY I = S2 · XIIII, XIY Y Y = S3 · XIIII, . . . are also ways of flipping only the first qubit !

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

If s = 010, what is E ? (that affects the least number of qubits) E = XIIII ↔ Only 1st qubit flipped. XXY II = S1 · XIIII, Y IIY I = S2 · XIIII, XIY Y Y = S3 · XIIII, . . . are also ways of flipping only the first qubit ! The errors E, E · S1, E · S2, E · S3, . . . where (Si ∈ S) are “equivalent”. (degenerate errors)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

If s = 010, what is E ? (that affects the least number of qubits) E = XIIII ↔ Only 1st qubit flipped. XXY II = S1 · XIIII, Y IIY I = S2 · XIIII, XIY Y Y = S3 · XIIII, . . . are also ways of flipping only the first qubit ! The errors E, E · S1, E · S2, E · S3, . . . where (Si ∈ S) are “equivalent”. (degenerate errors) Pr(flipping first qubit) = Pr(E)+Pr(E ·S1)+Pr(E ·S2)+· · · =

S∈S Pr(E ·S) ≡ Pr([E])

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

Problem of interest: Degenerate Quantum Maximum likelihood decoding (DQMLD) DQMLD: Given the graph and s find the class [E] that has the maximum probability sum.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

Problem of interest: Degenerate Quantum Maximum likelihood decoding (DQMLD) DQMLD: Given the graph and s find the class [E] that has the maximum probability sum. There are many errors for a syndrome s with different probabilities:

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

Problem of interest: Degenerate Quantum Maximum likelihood decoding (DQMLD) DQMLD: Given the graph and s find the class [E] that has the maximum probability sum. There are many errors for a syndrome s with different probabilities: Classical decoding strategy → Find the maximum probable error

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

Problem of interest: Degenerate Quantum Maximum likelihood decoding (DQMLD) DQMLD: Given the graph and s find the class [E] that has the maximum probability sum. There are many errors for a syndrome s with different probabilities: Quantum → Group into classes and then find the maximum → harder in the quantum case

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Decoding Stabilizer codes

Problem of interest: Degenerate Quantum Maximum likelihood decoding (DQMLD) DQMLD: Given the graph and s find the class [E] that has the maximum probability sum. There are many errors for a syndrome s with different probabilities: Special case: Large “gap” (∆) between maximum sum and others (Classical decoding)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Contents of this talk

1 Computational Complexity 2 Classical error correction 3 Quantum error correction 4 Main result 5 Conclusions

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Our main result

Decoding a quantum stabilizer code is #P-Complete. (Informal statement)

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Our main result

Decoding a quantum stabilizer code is #P-Complete. (Informal statement) For a graph with n qubits ’s and n − k checks ’s, . . . Main result: Hardness of DQMLD DQMLD on [[n, k = 1]] stabilizer code on an independent X − Z channel and with a promise gap ∆ ≤ 2[2 + nλ]−1, with λ = Ω(polylog(n)), is in #P-Complete.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Our main result

Decoding a quantum stabilizer code is #P-Complete. (Informal statement) For a graph with n qubits ’s and n − k checks ’s, . . . Main result: Hardness of DQMLD DQMLD on [[n, k = 1]] stabilizer code on an independent X − Z channel and with a promise gap ∆ ≤ 2[2 + nλ]−1, with λ = Ω(polylog(n)), is in #P-Complete. The proof outline: Weight enumerator problem ≤p DQMLD proves DQMLD ∈ #P-Complete.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Contents of this talk

1 Computational Complexity 2 Classical error correction 3 Quantum error correction 4 Main result 5 Conclusions

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Conclusion and references . . .

It was long known that decoding classical codes is hard . . . (NP-Complete)

• E. Berlekamp, R. McEliece, and H. Van Tilborg, “On the inherent intractability of

certain coding problems (corresp.)” IEEE Transactions on Information Theory, vol. 24,

• pp. 384 – 386, 1978.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Conclusion and references . . .

It was long known that decoding classical codes is hard . . . (NP-Complete)

• E. Berlekamp, R. McEliece, and H. Van Tilborg, “On the inherent intractability of

certain coding problems (corresp.)” IEEE Transactions on Information Theory, vol. 24,

• pp. 384 – 386, 1978.

It is known that decoding a quantum stabilizer code is at least as hard (NP-Complete) M.H. Hsieh and F. Le Gall, “NP-hardness of decoding quantum error-correction codes” Phys. Rev. A, vol. 83, p. 052331, 2011.

Pavithran Iyer Hardness of decoding stabilizer codes

Computational Complexity Classical error correction Quantum error correction Main result Conclusions

Conclusion and references . . .

It was long known that decoding classical codes is hard . . . (NP-Complete)

• E. Berlekamp, R. McEliece, and H. Van Tilborg, “On the inherent intractability of

certain coding problems (corresp.)” IEEE Transactions on Information Theory, vol. 24,

• pp. 384 – 386, 1978.

It is known that decoding a quantum stabilizer code is at least as hard (NP-Complete) M.H. Hsieh and F. Le Gall, “NP-hardness of decoding quantum error-correction codes” Phys. Rev. A, vol. 83, p. 052331, 2011. We show that decoding a stabilizer code is much harder (#P-Complete) Pavithran Iyer and David Poulin “Hardness of decoding quantum stabilizer codes” arXiv:1310.3235.

Pavithran Iyer Hardness of decoding stabilizer codes