# shor's algorithm qiskit

Multiple Qubits and Entangled States, 2.3 You should try re-running the cell a few times to see how it behaves. Using a quantum computer to factor the extremely large numbers used in RSA is decades away and will require an error-corrected device with many qubits— but today, we can at least use it to factor very small coprimes…like 15. Defining Quantum Circuits, 3.2 After the final measurement of register 1 in step 9 we obtain some integer m, which has a high probability of being an integer multiple of q/r. Calibrating Qubits with Qiskit Pulse, 6.2 Curious, you read the contents of the slip: At the bottom, you see what you can only assume is the coprime of an RSA key, , 15). The benefit of quantum computing posits that they can solve real-world problems more efficiently then classical computers. Overview of Shor's Algorithm. # Add these values to the rows in our table: # Get fraction that most closely resembles 0.666, """Compute a^{2^j} (mod N) by repeated squaring""", # This is to make sure we get reproduceable results, # Initialise counting qubits in state |+>, # Setting memory=True below allows us to see a list of each sequential reading, # Denominator should (hopefully!) Quickly, you use the factors P and Q to restore the incomplete private key. Since the best-known classical algorithm requires superpolynomial time to factor the product of two primes, the widely used cryptosystem, RSA, relies on factoring being impossible for large enough integers. Python has this functionality built in: We can use the fractions module to turn a float into a Fraction object, for example: Because this gives fractions that return the result exactly (in this case, 0.6660000...), this can give gnarly results like the one above. and return possible exponents for period finding. For example with $a = 3$ and $N = 35$: So a superposition of the states in this cycle ($|u_0\rangle$) would be an eigenstate of $U$: This eigenstate has an eigenvalue of 1, which isnât very interesting. When calculating the unitary gate for amodN, the textbook uses the following for N=5 but doesn't provide an explanation as to why 3. 2) 11–11:15 PM — Note Comparison. I am trying to follow along with shor's algorithm. In this case, α will be less than log 2 N. Thus we can basically try all possible α’s with only linear overhead. Well, that didn’t work — RSA is too secure to simply be guessed. After all the work done in the previous posts, we are now ready to actually implement Shor’s factoring algorithm on a real quantum computer, using once more IBMs Q Experience and the Qiskit framework. Introduction to Quantum Error Correction using Repetition Codes, 5.2 Bernstein-Vazirani Algorithm, 3.6 So we got the motivation to develop an algorithm for period finding and the benefit of using QFT for this algorithm (naturally every engineer knows that FFT is used for finding frequencies, so it is a natural step) .Now let’s combine the packet. We provide the circuits for $U$ where: without explanation. Introduction, 1.2 This is when you connect to your quantum computer and begin your period-finding circuit. For example in this paper the number 15 is factored using only 5 qubits. A more interesting eigenstate could be one in which the phase is different for each of these computational basis states. Check it out: To give a better sense of how this algorithm might work in the real world, Qiskit Advocate Spencer Churchill imagined what might happen if you found RSA-encrypted code in the real world, and how Shor’s algorithm would be able to crack it. Shor’s Algorithm Watch Party. Now, a number a between 1 and n exclusive is randomly picked. We’re going through uncertain times. Actually there is an eﬃcient classical algorithm for this case. This tutorial will use a basic form of RSA to highlight the capability of Shor’s algorithm. For quantum announcements, updates, and general banter. The quantum Fourier transform is a key building block of many quantum algorithms, from Shor’s factoring algorithm over matrix inversion to quantum phase estimation and simulations.Time to see how this can be implemented with Qiskit. Qiskit Slack. Recall that the quantum Fourier transform (or, depending on conventions, its inverse) is given by Simulating Molecules using VQE, 4.1.3 From Qubit to Shor’s Algorithm. It was invented in 1994 by the American mathematician Peter Shor. then decrypt the listing with the private key. Implementations of Recent Quantum Algorithms, 4.2.1 Experimenting with Quantum Computing at IBM Qiskit Global Summer School 2020 August 9, 2020 sigmoid. In total you need 4n + 2 qubits to run Shor's algorithm.. Simon's algorithm, first introduced in Reference [1], was the first quantum algorithm to show an exponential speed-up versus the best classical algorithm in solving a specific problem. The security of many present-day cryptosystems is based on the assumption that no fast algorithm exists for factoring. Well for starters, Shor's Algorithm is an algorithm designed to be run on a quantum computer. 2.Pick a random integer x, are all being initialized with Hadamard (H) gates and the target qubits are being initialized at |1>. For this demonstration we will provide the circuits for $U$ without explanation, but in section 4 we will discuss how circuits for $U^{2^j}$ can be constructed efficiently. First, recall that Shor’s algorithm is designed to factor an integer M, with the restriction that M is supposed to be odd and not a prime power. Now, onto our ulterior goal of factoring, we first check if the number is even or of the form a b before using Shor’s algorithm, but we know that we are dealing with large prime numbers, so let’s jump onto that case. Third, you perform an inverse quantum Fourier transform on the measurement qubits. If N is even, return the factor 2. Proving Universality, 2.6 Quantum Teleportation, 3.3 Shor's Algorithm, 3.10 Dies stellt beispielsweise eine Gefahr für die häufig zur verschlüsselten Datenübertragung verwendeten RSA-Kryptosysteme dar, deren Sicherheit gerade auf der Annahme beruht, dass kein Faktorisierungsverfahr… Shor’s Algorithm is a quantum algorithm for integer factorisation. The Atoms of Computation, 1.3 Shorâs algorithm is famous for factoring integers in polynomial time. 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: 1. See you soon.”. Introduction, 2.2 Classical Computation on a Quantum Computer, 3. from qiskit.aqua.algorithms import Shor a, N = 2, 3 shor = Shor(N, a) circuit = shor.construct_circuit() print(circuit.draw()) # or circuit.draw(output='mpl') for a nicer looking diagram ;) Warning: the circuit is huge and takes a long time to plot for large N ! Luckily, you attended Abe’s lecture on Shor’s algorithm and know exactly where to begin! Shorâs solution was to use quantum phase estimation on the unitary operator: To see how this is helpful, letâs work out what an eigenstate of U might look like. In Shor's algorithm, you perform the QFT in such a manner that the entire answer is given to you at once. Shor's algorithm is a polynomial-time quantum computer algorithm for integer factorization. Hamiltonian Tomography, 7. The Case for Quantum, 2. Two distinct pieces of information are required to obtain the full range of the RSA function, a public and a private key. There was some work done on lowering the qubit requirements. Measuring the Qubit ac-Stark Shift, 6.7 For now its enough to show that if we can compute the period of $a^x\bmod N$ efficiently, then we can also efficiently factor. Very conveniently, if we sum up all these eigenstates, the different phases cancel out all computational basis states except $|1\rangle$: Since the computational basis state $|1\rangle$ is a superposition of these eigenstates, which means if we do QPE on $U$ using the state $|1\rangle$, we will measure a phase: Where $s$ is a random integer between $0$ and $r-1$. This algorithm is the source of much interest in the quantum community — one day perhaps a few decades in the future, these devices would be able to use Shor’s algorithm to crack RSA, the encryption that safeguards much of our data. Quantum Key Distribution, 4. The asymmetric modular function is constructed in such a way as to only allow the private key to unlock the encryption. Join the workspace #general. To see an example of factoring on a small number of qubits, we will factor 15, which we all know is the product of the not-so-large prime numbers 3 and 5. I spent two weeks of my quarantine having fun and learning so much from the many lecturers, mentors, and peers contributing on Crowdcast and Discord. Simply put given an odd integer N it will find it’s prime factors. Shor’s algorithm is famous for factoring integers in polynomial time. Shor’s algorithm is arguably the most dramatic example of how the paradigm of quantum computing changed our perception of which problems should be considered tractable. Work with a ﬁxed α. And if $r$ is also even, then we can write: (if $r$ is not even, we cannot go further and must try again with a different value for $a$). Setting Up Your Environment, 0.2 Quantum Fourier Transform, 3.8 Representing Qubit States, 1.4 Shor's algorithm is a manifestation of QC's advantage over classical computers. How do we do this? It was the summer of 2020; well, monsoon time in the southern part of India. The proliferation of noisy intermediate-scale quantum (NISQ) devices has allowed interested individuals to discover and develop scalable applications of quantum computing (QC). Since we aim to focus on the quantum part of the algorithm, we will jump straight to the case in which N is the product of two primes. First compute log 2 N with enough precision (what this eﬃciency needs to be will be clear from the rest of the algorithm.) ... 10–11 PM — 1 hour of the video by Qiskit Qummer School. Since the best-known classical algorithm requires superpolynomial time to factor the product of two primes, the widely used cryptosystem, RSA, relies on factoring being impossible for large enough integers. These bad results are because $s = 0$, or because $s$ and $r$ are not coprime and instead of $r$ we are given a factor of $r$. Hybrid quantum-classical Neural Networks with PyTorch and Qiskit, 4.2 Shor's algorithm at the "Period-finding subroutine" uses two registers, possibly as big as 2n + 1 where n is number of bits needed to represent the number to factor. Using RSA and Shor's Algorithm, you determine the private key to be: You learn that the decrypted listing is IBM! The following code is Shor's algorithm in Python: Quantum Phase Estimation, 3.9 The part I am having trouble with is the operators at the bottom. Merged Shor's Algorithm Tutorial #131. By the fourth day, we were assigned a lab factoring the coprime 15. Unfortunately, despite scaling polynomially with $j$, modular exponentiation circuits are not straightforward and are the bottleneck in Shorâs algorithm. As the algorithm runs the state of the quantum memory register changes in the manner laid out in the description of Shor's algorithm. The easiest solution to this is to simply repeat the experiment until we get a satisfying result for $r$. Linear Algebra, 8.2 If you want to learn more about the machinery that goes into Shor’s algorithm, namely Quantum Phase Estimation and the Quantum Fourier Transform, check out lectures 7 through 9 on our Introduction to Quantum Computing and Quantum Hardware course. The rst improvement, as hinted before, is that when adding a number that is known classically at compile time, the addition can be reduced to unitary single qubit gates in … What results do you get and why. If we started in the state $|1\rangle$, we can see that each successive application of U will multiply the state of our register by $a \pmod N$, and after $r$ applications we will arrive at the state $|1\rangle$ again. What better way to spend time than to learn about uncertainties? Stephane Beauregard, Circuit for Shor's algorithm using 2n+3 qubits, arXiv:quant-ph/0205095, M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge Series on Information and the Natural Sciences (Cambridge University Press, Cambridge, 2000). Quantum computers much like classical ones can with n bits present 2^n different values. This is not the only eigenstate with this behaviour; to generalise this further, we can multiply an integer, $s$, to this phase difference, which will show up in our eigenvalue: We now have a unique eigenstate for each integer value of $s$ where $$0 \leq s \leq r-1$$. Variational Quantum Linear Solver, 5. Second, you see U gates applying a unitary operator, U(x) = a^x (mod N), on the target qubits controlled by the measurement qubits, which in your case is. Remember that the phase we measure will be $s/r$ where: and $s$ is a random integer between 0 and $r-1$. We want a way to create the operator: that grows polynomially with $j$. In fact, there are specific criteria for choosing numbers that are difficult to factor, but the basic idea is to choose the product of two large prime numbers. A company is going to report high earnings. Basic Synthesis of Single-Qubit Gates, 8.1 Measurement Error Mitigation, 5.3 Accessing Higher Energy States, 6.3 Classical Part. In our case, since we are only dealing with exponentials of the form $2^j$, the repeated squaring algorithm becomes very simple: If an efficient algorithm is possible in Python, then we can use the same algorithm on a quantum computer. ', """Controlled multiplication by a mod 15""", """n-qubit QFTdagger the first n qubits in circ""", # Create QuantumCircuit with n_count counting qubits. In this example we will solve the period finding problem for $a=7$ and $N=15$. Quantum Counting, 3.12 Circuit Quantum Electrodynamics, 6.5 Simon's Algorithm, 3.7 You look up to see a man hastily exit the New York City subway, leaving behind a scrap of paper on the floor. ), before using Shorâs period finding for the worst-case scenario. [1] Informally, it solves… However, only people with the actual prime numbers themselves can decrypt the data; this is called the private key. Introduction. Since a factoring problem can be turned into a period finding problem in polynomial time, an efficient period finding algorithm can be used to factor integers efficiently too. Shor's algorithm hinges on a result from number theory. The functions below simply use the properties of asymmetric algorithms to encode and decode text using public and private keys. Fortunately, calculating: efficiently is possible. Randomized Benchmarking, 5.4 The period, or order ($r$), is the smallest (non-zero) integer such that: We can see an example of this function plotted on the graph below. Note that the lines between points are to help see the periodicity and do not represent the intermediate values between the x-markers. You may have noticed that the method of creating the $U^{2^j}$ gates by repeating $U$ grows exponentially with $j$ and will not result in a polynomial time algorithm. Now that we have a refresher on what Q-bits are lets take a look at how they can be helpful. Now, you sort through the possible exponents, finding those which satisfy two constraints: Using an applicable period, x, you can find nontrivial factors, P and Q , of N with gcd(a^(x/2) ± 1, N) . Manner that the entire answer is given to you at once best way to implement it on! A number a between 1 and N exclusive is randomly picked a value generated for you day... It was invented in 1994 by the fourth day, we were assigned lab! Summer of 2020 ; well, monsoon time in the southern part of Shorâs algorithm which! I encrypted the listing would be to suggestions as to only allow the key. 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