What is a use case of factorization in quantum computing

Quantum computing could revolutionise the way we approach calculations—but can it really break modern encryption? What is factorization? Factorization is the process where a number is written as a product of smaller numbers. For instance, consider the number 24. 24 can be represented by 1x24, 2x12, 3x8, or 4x6. 24 is rather a small number, so writing down all possible factorizations is straightforward. But for larger numbers, how can we know that we have found all of the possibilities? One option is to continue factoring until all of the factors are prime. That is, each number has no factors other than 1 and itself. In our case, 24=2x2x2x3. All possible factorizations can be constructed from this list of prime numbers. For example, 24 = (2x2)x(2x3) = 4x6. The prime factorization of a number is like its fingerprint – it is unique to each one. A difficult problem What if we wanted to factor 49,189,447? This might take some time, because it can only be written as a product of two prime numbers: 6221x7907. Surely, a classical computer would be able to tackle this problem by trying a list of possibilities: 2, 3, 5, 7, and so on, until it found the correct numbers. But there are infinite prime numbers! The largest found to date (in 2018) has 24,862,048 digits when written down. If someone were to multiply this prime number with another massive one, even modern supercomputers wouldn’t be able to find the 2 factors. There’s something peculiar about the asymmetry of this problem. M...

Shor's algorithm is a N in N)3) time and O(log N) space, named after The algorithm is significant because it implies that N which is the product of two large prime numbers. One way to crack RSA encryption is by factoring N, but with classical algorithms, factoring becomes increasingly time-consuming as N grows large; more specifically, no classical algorithm is known that can factor in time O((log N) k) for any k. By contrast, Shor's algorithm can crack RSA in polynomial time. It has also been extended to attack many other public key cryptosystems. Like all quantum computer algorithms, Shor's algorithm is probabilistic: it gives the correct answer with high probability, and the probability of failure can be decreased by repeating the algorithm. Shor's algorithm was demonstrated in 2001 by a group at IBM, which factored 15 into 3 and 5, using a quantum computer with 7 Procedure The problem we are trying to solve is that, given an integer N, we try to find another integer p between 1 and N that divides N. Shor's algorithm consists of two parts: • A reduction of the factoring problem to the problem of • A quantum algorithm to solve the order-finding problem. Classical part • Pick a pseudo-random number a< N • Compute a, N). This may be done using the • If gcd( a, N) ≠ 1, then there is a nontrivial factor of N, so we are done. • Otherwise, use the period-finding subroutine (below) to find r, the f( x) = a x mod N , i.e. the smallest integer r for which f( x + r) = f( x). • If ...

• العربية • Български • Català • Dansk • Deutsch • Español • فارسی • Français • 한국어 • हिन्दी • Italiano • עברית • Lietuvių • Lombard • Magyar • Nederlands • Polski • Português • Русский • Simple English • Српски / srpski • Srpskohrvatski / српскохрватски • Suomi • ไทย • Türkçe • Українська • Tiếng Việt • 中文 This article includes a list of general it lacks sufficient corresponding Please help to ( September 2010) ( Shor's algorithm is a On a quantum computer, to factor an integer N . This can be factored using N ∣ ( a r / 2 − 1 ) ( a r / 2 + 1 ) . We compute d = gcd ( N , a r / 2 − 1 ) . • Pick a random number 1 < a < N , and we're done. Otherwise, go back to step 1. It has been shown that this will be likely to succeed after a few runs [ citation needed]. Quantum order-finding subroutine [ ] The quantum subroutine of Shor's algorithm can be expressed as an application of N into a state close to the associated eigenvalue. For the purposes of quantum order-finding, we employ this strategy using the unitary defined by the action: U | k ⟩ = : | ψ j ⟩ = 1 r ( | 1 ⟩ + ω r − j | a ⟩ + ω r − 2 j | a 2 ⟩ + … + ω r − j ( r − 1 ) | a r − 1 ⟩ ) state. This can be seen from the following: 1 r ∑ j = 0 r − 1 | ψ j ⟩ = 1 r ∑ j = 0 r − 1 ∑ k = 0 r − 1 ω r j k | a k ⟩ = 1 r ∑ k = 0 r − 1 ( ∑ j = 0 r − 1 ω r j k ) | a k ⟩ = | 1 ⟩ + 1 r ∑ k = 1 r − 1 ( ∑ j = 0 r − 1 ω r j k ) | a k ⟩ , since the roots of unity sum to zero: ∑ j = 0 r − 1 ω r j k = 0 would apply the transformation | 0 ...

Although the environmental effects, i.e., dissipation and decoherence seem to be the strongest adversaries in the quantum information realm, here, we address how dissipation can be harnessed for quantum state preparation and universal quantum computation. In this line, we propose a realistic scheme for hybrid classical-quantum neural networks based on dissipative two-qubit channels. In particular, we design a variational quantum circuit consisting of a set of universal quantum gates. We encode classical information in the initial states of a two-qubit system interacting with a global environment. This composite system plays the role of a dissipative quantum channel (DQC). A pooling layer concatenates the output states of the DQCs resulting in the outcome of the circuit. Both the DCQs and the pooling layer provide superposition and entanglement which are the key ingredients of any universal quantum computation protocol. Finally, we investigate the capability and adaptability of this model by doing some machine learning tasks. It is reasonable to postulate that a quantum computer based on DQCs may outperform a classical computer because, in contrast to the latter, the former is capable of producing atypical patterns through non-classical phenomena. The major challenge in the implementation of a quantum computer lies in the difficulty to maintain coherence to simulate unitary dynamics. Naturally, real physical systems interact with their environments and undergo the effects o...

Quantum computing could revolutionise the way we approach calculations—but can it really break modern encryption? What is factorization? Factorization is the process where a number is written as a product of smaller numbers. For instance, consider the number 24. 24 can be represented by 1x24, 2x12, 3x8, or 4x6. 24 is rather a small number, so writing down all possible factorizations is straightforward. But for larger numbers, how can we know that we have found all of the possibilities? One option is to continue factoring until all of the factors are prime. That is, each number has no factors other than 1 and itself. In our case, 24=2x2x2x3. All possible factorizations can be constructed from this list of prime numbers. For example, 24 = (2x2)x(2x3) = 4x6. The prime factorization of a number is like its fingerprint – it is unique to each one. A difficult problem What if we wanted to factor 49,189,447? This might take some time, because it can only be written as a product of two prime numbers: 6221x7907. Surely, a classical computer would be able to tackle this problem by trying a list of possibilities: 2, 3, 5, 7, and so on, until it found the correct numbers. But there are infinite prime numbers! The largest found to date (in 2018) has 24,862,048 digits when written down. If someone were to multiply this prime number with another massive one, even modern supercomputers wouldn’t be able to find the 2 factors. There’s something peculiar about the asymmetry of this problem. M...

• العربية • Български • Català • Dansk • Deutsch • Español • فارسی • Français • 한국어 • हिन्दी • Italiano • עברית • Lietuvių • Lombard • Magyar • Nederlands • Polski • Português • Русский • Simple English • Српски / srpski • Srpskohrvatski / српскохрватски • Suomi • ไทย • Türkçe • Українська • Tiếng Việt • 中文 This article includes a list of general it lacks sufficient corresponding Please help to ( September 2010) ( Shor's algorithm is a On a quantum computer, to factor an integer N . This can be factored using N ∣ ( a r / 2 − 1 ) ( a r / 2 + 1 ) . We compute d = gcd ( N , a r / 2 − 1 ) . • Pick a random number 1 < a < N , and we're done. Otherwise, go back to step 1. It has been shown that this will be likely to succeed after a few runs [ citation needed]. Quantum order-finding subroutine [ ] The quantum subroutine of Shor's algorithm can be expressed as an application of N into a state close to the associated eigenvalue. For the purposes of quantum order-finding, we employ this strategy using the unitary defined by the action: U | k ⟩ = : | ψ j ⟩ = 1 r ( | 1 ⟩ + ω r − j | a ⟩ + ω r − 2 j | a 2 ⟩ + … + ω r − j ( r − 1 ) | a r − 1 ⟩ ) state. This can be seen from the following: 1 r ∑ j = 0 r − 1 | ψ j ⟩ = 1 r ∑ j = 0 r − 1 ∑ k = 0 r − 1 ω r j k | a k ⟩ = 1 r ∑ k = 0 r − 1 ( ∑ j = 0 r − 1 ω r j k ) | a k ⟩ = | 1 ⟩ + 1 r ∑ k = 1 r − 1 ( ∑ j = 0 r − 1 ω r j k ) | a k ⟩ , since the roots of unity sum to zero: ∑ j = 0 r − 1 ω r j k = 0 would apply the transformation | 0 ...

Shor's algorithm is a N in N)3) time and O(log N) space, named after The algorithm is significant because it implies that N which is the product of two large prime numbers. One way to crack RSA encryption is by factoring N, but with classical algorithms, factoring becomes increasingly time-consuming as N grows large; more specifically, no classical algorithm is known that can factor in time O((log N) k) for any k. By contrast, Shor's algorithm can crack RSA in polynomial time. It has also been extended to attack many other public key cryptosystems. Like all quantum computer algorithms, Shor's algorithm is probabilistic: it gives the correct answer with high probability, and the probability of failure can be decreased by repeating the algorithm. Shor's algorithm was demonstrated in 2001 by a group at IBM, which factored 15 into 3 and 5, using a quantum computer with 7 Procedure The problem we are trying to solve is that, given an integer N, we try to find another integer p between 1 and N that divides N. Shor's algorithm consists of two parts: • A reduction of the factoring problem to the problem of • A quantum algorithm to solve the order-finding problem. Classical part • Pick a pseudo-random number a< N • Compute a, N). This may be done using the • If gcd( a, N) ≠ 1, then there is a nontrivial factor of N, so we are done. • Otherwise, use the period-finding subroutine (below) to find r, the f( x) = a x mod N , i.e. the smallest integer r for which f( x + r) = f( x). • If ...

Although the environmental effects, i.e., dissipation and decoherence seem to be the strongest adversaries in the quantum information realm, here, we address how dissipation can be harnessed for quantum state preparation and universal quantum computation. In this line, we propose a realistic scheme for hybrid classical-quantum neural networks based on dissipative two-qubit channels. In particular, we design a variational quantum circuit consisting of a set of universal quantum gates. We encode classical information in the initial states of a two-qubit system interacting with a global environment. This composite system plays the role of a dissipative quantum channel (DQC). A pooling layer concatenates the output states of the DQCs resulting in the outcome of the circuit. Both the DCQs and the pooling layer provide superposition and entanglement which are the key ingredients of any universal quantum computation protocol. Finally, we investigate the capability and adaptability of this model by doing some machine learning tasks. It is reasonable to postulate that a quantum computer based on DQCs may outperform a classical computer because, in contrast to the latter, the former is capable of producing atypical patterns through non-classical phenomena. The major challenge in the implementation of a quantum computer lies in the difficulty to maintain coherence to simulate unitary dynamics. Naturally, real physical systems interact with their environments and undergo the effects o...

• العربية • Български • Català • Dansk • Deutsch • Español • فارسی • Français • 한국어 • हिन्दी • Italiano • עברית • Lietuvių • Lombard • Magyar • Nederlands • Polski • Português • Русский • Simple English • Српски / srpski • Srpskohrvatski / српскохрватски • Suomi • ไทย • Türkçe • Українська • Tiếng Việt • 中文 This article includes a list of general it lacks sufficient corresponding Please help to ( September 2010) ( Shor's algorithm is a On a quantum computer, to factor an integer N . This can be factored using N ∣ ( a r / 2 − 1 ) ( a r / 2 + 1 ) . We compute d = gcd ( N , a r / 2 − 1 ) . • Pick a random number 1 < a < N , and we're done. Otherwise, go back to step 1. It has been shown that this will be likely to succeed after a few runs [ citation needed]. Quantum order-finding subroutine [ ] The quantum subroutine of Shor's algorithm can be expressed as an application of N into a state close to the associated eigenvalue. For the purposes of quantum order-finding, we employ this strategy using the unitary defined by the action: U | k ⟩ = : | ψ j ⟩ = 1 r ( | 1 ⟩ + ω r − j | a ⟩ + ω r − 2 j | a 2 ⟩ + … + ω r − j ( r − 1 ) | a r − 1 ⟩ ) state. This can be seen from the following: 1 r ∑ j = 0 r − 1 | ψ j ⟩ = 1 r ∑ j = 0 r − 1 ∑ k = 0 r − 1 ω r j k | a k ⟩ = 1 r ∑ k = 0 r − 1 ( ∑ j = 0 r − 1 ω r j k ) | a k ⟩ = | 1 ⟩ + 1 r ∑ k = 1 r − 1 ( ∑ j = 0 r − 1 ω r j k ) | a k ⟩ , since the roots of unity sum to zero: ∑ j = 0 r − 1 ω r j k = 0 would apply the transformation | 0 ...

Shor's algorithm is a N in N)3) time and O(log N) space, named after The algorithm is significant because it implies that N which is the product of two large prime numbers. One way to crack RSA encryption is by factoring N, but with classical algorithms, factoring becomes increasingly time-consuming as N grows large; more specifically, no classical algorithm is known that can factor in time O((log N) k) for any k. By contrast, Shor's algorithm can crack RSA in polynomial time. It has also been extended to attack many other public key cryptosystems. Like all quantum computer algorithms, Shor's algorithm is probabilistic: it gives the correct answer with high probability, and the probability of failure can be decreased by repeating the algorithm. Shor's algorithm was demonstrated in 2001 by a group at IBM, which factored 15 into 3 and 5, using a quantum computer with 7 Procedure The problem we are trying to solve is that, given an integer N, we try to find another integer p between 1 and N that divides N. Shor's algorithm consists of two parts: • A reduction of the factoring problem to the problem of • A quantum algorithm to solve the order-finding problem. Classical part • Pick a pseudo-random number a< N • Compute a, N). This may be done using the • If gcd( a, N) ≠ 1, then there is a nontrivial factor of N, so we are done. • Otherwise, use the period-finding subroutine (below) to find r, the f( x) = a x mod N , i.e. the smallest integer r for which f( x + r) = f( x). • If ...