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 5 th − 6 th , 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. A B Consider a simple code: C = { 000 , 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. A B Consider a simple code: C = { 000 , 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. A B Consider a simple code: C = { 000 , 111 } . If � r = 001 is received → some bit(s) were flipped. which ones ? ↔ what was added ? e ) ∼ p 2 � e = 001 ↔ Last bit flipped: Pr ( � e ) ∼ p � e = 110 ↔ first two bits flipped: Pr ( � 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
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