What is a use case of factorization in quantum computing?

beginner here. I've always heard explanations on quantum computing, all about superposition, entanglement, etc. But how does superposition actually apply to quantum computing? Yeah, its "in between" a 1 and a 0 but why does this matter? How do quantum computers actually use this feature, and why is it so important? A quantum computer is computing device that makes use of quantum state instead of classical states. A quantum state, also known as a state vector, contains statistical information about the quantum system. It’s essentially a probability density. Quantum states can have interesting properties like superposition, entanglement, and interference effects. Now, a bit is the building block of classical computers, and it can be in the states 0 or 1, a classical two level system. And a qubit is the building block of quantum computers, it is a two-level quantum system, hence it is a unit vector in the space $\mathbb |\alpha|^2 + |\beta|^2 = 1 $$ Note how the state of the qubit is different than that of a classical bit. It can be written in a linear combination or superposition between the state $|0\rangle$ and the state $|1\rangle$. This is fine because we must remember that a qubit, after all, is a quantum mechanical object, it inherently possess quantum mechanical properties like superposition. So if you have a quantum computer, it must be able to deal with superposition of states, because if it can't then it is not really a quantum computer. How does a superposition pl...

• 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 ...

Summary. Digital computing has limitations in regards to an important category of calculation called combinatorics, in which the order of data is important to the optimal solution. These complex, iterative calculations can take even the fastest computers a long time to process. Computers and software that are predicated on the assumptions of quantum mechanics have the potential to perform combinatorics and other calculations much faster, and as a result many firms are already exploring the technology, whose known and probable applications already include cybersecurity, bio-engineering, AI, finance, and complex manufacturing. Quantum technology is approaching the mainstream. Goldman Sachs To understand what’s going on, it’s useful to take a step back and examine what exactly it is that computers do. Let’s start with today’s digital technology. At its core, the digital computer is an arithmetic machine. It made performing mathematical calculations cheap and its impact on society has been immense. Advances in both hardware and software have made possible the application of all sorts of computing to products and services. Today’s cars, dishwashers, and boilers all have some kind of computer embedded in them — and that’s before we even get to smartphones and the internet. Without computers we would never have reached the moon or put satellites in orbit. These computers use binary signals (the famous 1s and 0s of code) that are measured in “bits” or bytes. The more complicated t...

According to the According to the New Scientist: A quantum computing start-up company called Zapata has worked with IBM to develop a new way to factor large numbers, using it on the largest number that has been factored with a quantum computer so far... The future success of the algorithm used could have big implications • What is this new algorithm? • How many q-bits are required to factor 1,099,551,473,989? $\begingroup$ To put the ability to factor 1099551473989 into perspective, GNU factor command included in the Linux coreutils package is able to factor that number into correct factors in about 0.8 ms including the overhead of starting the process and about 0.55 ms of CPU time when executed on nearly decade old i5-3570K. So "state of art" quantum computer can do about the same that a single core of decade old desktop computer can do in less than 1/1000 of a second. You can literally enter factor 1099551473989 on any Linux terminal and you get the results faster than your display can render it. $\endgroup$ The claimed new quantum factoring record is $n=a(a+b)$ with $a=1048589=2^+\mathcal o(1/\sqrt n)$. [#] Auxiliary proof: Assume $a(a+b)=a'(a'+b')$ with $0\le b\le12$ and $0\le b'\le12$. Let $c=2a-b$ and $c'=2a'-b'$. It comes $(c-b)(c+b)/4=(c'-b')(c'+b')/4$, thus $c^2-b^2=c'^2-b'^2$, thus $c^2-c'^2=b^2-b'^2$, thus $|(c-c')(c+c')|\le144$, thus for $c\ge72$ and $c'\ge72$ the only solution is $c=c'$, hence $b=b'$. Thus for $a\ge42$ and $a'\ge42$ the only solution is $a=a'$...

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...

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...

Summary. Digital computing has limitations in regards to an important category of calculation called combinatorics, in which the order of data is important to the optimal solution. These complex, iterative calculations can take even the fastest computers a long time to process. Computers and software that are predicated on the assumptions of quantum mechanics have the potential to perform combinatorics and other calculations much faster, and as a result many firms are already exploring the technology, whose known and probable applications already include cybersecurity, bio-engineering, AI, finance, and complex manufacturing. Quantum technology is approaching the mainstream. Goldman Sachs To understand what’s going on, it’s useful to take a step back and examine what exactly it is that computers do. Let’s start with today’s digital technology. At its core, the digital computer is an arithmetic machine. It made performing mathematical calculations cheap and its impact on society has been immense. Advances in both hardware and software have made possible the application of all sorts of computing to products and services. Today’s cars, dishwashers, and boilers all have some kind of computer embedded in them — and that’s before we even get to smartphones and the internet. Without computers we would never have reached the moon or put satellites in orbit. These computers use binary signals (the famous 1s and 0s of code) that are measured in “bits” or bytes. The more complicated t...

beginner here. I've always heard explanations on quantum computing, all about superposition, entanglement, etc. But how does superposition actually apply to quantum computing? Yeah, its "in between" a 1 and a 0 but why does this matter? How do quantum computers actually use this feature, and why is it so important? A quantum computer is computing device that makes use of quantum state instead of classical states. A quantum state, also known as a state vector, contains statistical information about the quantum system. It’s essentially a probability density. Quantum states can have interesting properties like superposition, entanglement, and interference effects. Now, a bit is the building block of classical computers, and it can be in the states 0 or 1, a classical two level system. And a qubit is the building block of quantum computers, it is a two-level quantum system, hence it is a unit vector in the space $\mathbb |\alpha|^2 + |\beta|^2 = 1 $$ Note how the state of the qubit is different than that of a classical bit. It can be written in a linear combination or superposition between the state $|0\rangle$ and the state $|1\rangle$. This is fine because we must remember that a qubit, after all, is a quantum mechanical object, it inherently possess quantum mechanical properties like superposition. So if you have a quantum computer, it must be able to deal with superposition of states, because if it can't then it is not really a quantum computer. How does a superposition pl...

• 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 ...

According to the According to the New Scientist: A quantum computing start-up company called Zapata has worked with IBM to develop a new way to factor large numbers, using it on the largest number that has been factored with a quantum computer so far... The future success of the algorithm used could have big implications • What is this new algorithm? • How many q-bits are required to factor 1,099,551,473,989? $\begingroup$ To put the ability to factor 1099551473989 into perspective, GNU factor command included in the Linux coreutils package is able to factor that number into correct factors in about 0.8 ms including the overhead of starting the process and about 0.55 ms of CPU time when executed on nearly decade old i5-3570K. So "state of art" quantum computer can do about the same that a single core of decade old desktop computer can do in less than 1/1000 of a second. You can literally enter factor 1099551473989 on any Linux terminal and you get the results faster than your display can render it. $\endgroup$ The claimed new quantum factoring record is $n=a(a+b)$ with $a=1048589=2^+\mathcal o(1/\sqrt n)$. [#] Auxiliary proof: Assume $a(a+b)=a'(a'+b')$ with $0\le b\le12$ and $0\le b'\le12$. Let $c=2a-b$ and $c'=2a'-b'$. It comes $(c-b)(c+b)/4=(c'-b')(c'+b')/4$, thus $c^2-b^2=c'^2-b'^2$, thus $c^2-c'^2=b^2-b'^2$, thus $|(c-c')(c+c')|\le144$, thus for $c\ge72$ and $c'\ge72$ the only solution is $c=c'$, hence $b=b'$. Thus for $a\ge42$ and $a'\ge42$ the only solution is $a=a'$...