Shors'algorithm simplified.pptx

Shors'algorithm simplified.pptx

• 1.
  Quantum Computation Simplified Kathiresan S Part- 6 Shor’s algorithm, QPE & QFT
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  Introduction What is Shor’salgorithm?  This is to find prime factors of an integer and invented by an American mathematician Peter Shor in 1994  Could be used to break RSA cryptography schemes (used in online transactions).  This is much faster than the most efficient known classical factoring algorithm. (eg. To break RSA 2048 which will use a 617-digit number, it will take many years in classical computer and a few seconds in a perfect Quantum Computer)
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  Prerequisite for thisvideo Essential  Basic knowledge of Quantum Computing – my previous videos (Part 1 to 5) or equivalent from other videos/ textbooks Desirable  Fourier transform  Prime factorization  Euclidean algorithm for finding GCD  Continued fraction  Root of unity
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  Factoring Algorithm  N= P.Q where N is an odd composite number and P & Q are odd prime numbers (of roughly equal in length/ value)  𝑋2 = 1(mod N): 𝑋 ≠ ±1 (mod N)  𝑋2 − 1 = 0(mod N)  N divides (𝑋2 − 1) or (X - 1)(X + 1) but not X - 1 and X + 1  GCD (N, X -1) and GCD(N, X + 1) will give factors of N  Pick x a random number and find its period r such that 𝑥𝑟 = 1 mod N , Finding Greatest Common Divisor (GCD) (𝑥𝑟/2 + 1, 𝑁) 𝑎𝑛𝑑/𝑜𝑟 GCD (𝑥𝑟/2 − 1, 𝑁) will give factors of N  Attempt fails when r is odd or 𝑥𝑟/2 = −1 (𝑚𝑜𝑑 𝑁). Repeat above step till these two conditions are not met.  Probability of guessing the right x is 1 − 1 2𝑘−1 where k is the number of distinct odd prime factors of N
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  How to getperiod ‘r’  What is period r? The period or order (r), is the smallest (non-zero) integer such that xr mod N = 1  Let us assume N = 15, x = 2 , 21(mod 15) = 2, 22(mod 15) = 4, 23(mod 15) =8, 24(mod 15) = 1, 25(mod 15) = 2, 26(mod 15) = 4, 27(mod 15) = 8,..  r can be found out using Quantum Phase Estimation algorithm  𝑈 𝜑 = 𝑒2𝜋𝑖𝜃 𝜑 where U is a unitary matrix and 𝑒2𝜋𝑖𝜃 is its eigen value and | 𝜑 is its eigen vector  In order to measure 𝜃, we need to know about Quantum Fourier Transform (& its inverse) and Continued Fraction
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  Quantum Fourier Transform(QFT)  Similar to FFT/ DFT, transforms between two bases, the computational (Z) basis, and the Fourier basis 8 (decimal) in Computational Basis 8 (decimal) in Fourier Basis Rotation of qubits – Multiples of 8 2𝑛 x 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 = 8 16 x 360 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 15 (decimal) in Computational Basis 15 (decimal) in Fourier Basis
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  Quantum Fourier Transform Discrete Fourier Transform DFT of a vector of complex numbers, x0,….,xN-1 is given by the following equation wherein yk = y0,…...yN-1 𝑦𝑘 ≡ 1 𝑁 𝑗=0 𝑁−1 𝑥𝑗𝑒2𝜋𝑖𝑗𝑘/𝑁  Quantum Fourier Transform QFT on an orthonormal basis states |0 , … … , |𝑁 − 1 is given by the following equation |𝑗 → 1 𝑁 𝑘=0 𝑁−1 𝑒2𝜋𝑖𝑗𝑘/𝑁|𝑘 Assuming N = 2n where n is an integer, |𝑗 → 1 2𝑛/2 𝑘=0 2𝑛−1 𝑒2𝜋𝑖𝑗𝑘/2𝑛 |𝑘
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  Quantum Fourier Transform Fromprevious slide, |𝑗 → 1 2𝑛/2 𝑘=0 2𝑛−1 𝑒2𝜋𝑖𝑗𝑘/2𝑛 |𝑘 Normally, we need to have a Unitary matrix to do this operation. It is given by, We will simplify the matrix with known small gates (unitary matrices) |𝑗1 … … … … . . 𝑗𝑛 → |0 + 𝑒2𝜋𝑖0.𝑗𝑛|1 |0 + 𝑒2𝜋𝑖0.𝑗𝑛−1𝑗𝑛|1 … … |0 + 𝑒2𝜋𝑖0.𝑗1𝑗2……𝑗𝑛|1 2𝑛/2
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  Quantum Fourier Transform Fromprevious slide, |𝑗1,……….𝑗𝑛 → |0 + 𝑒2𝜋𝑖0.𝑗𝑛|1 |0 + 𝑒2𝜋𝑖0.𝑗𝑛−1𝑗𝑛|1 … … |0 + 𝑒2𝜋𝑖0.𝑗1𝑗2……𝑗𝑛|1 2𝑛/2 𝑤ℎ𝑒𝑟𝑒𝑖𝑛 0. 𝑗1𝑗2 ….. 𝑗𝑛 = 𝑗1 21 + 𝑗2 22 + ⋯ … … + 𝑗𝑛 2𝑛 This can be achieved using three types of gates - Hadamard gate 1 2 1 1 1 −1 and a Rotational gate 𝑅𝑘 = 1 0 0 𝑒 2𝜋𝑖 2𝑘 𝑤ℎ𝑒𝑟𝑒 𝑒 2𝜋𝑖 2𝑘 is 2k th root of unity and the circuit will look like this (we need to have swap gates to maintain the order)
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  Quantum Fourier Transform Letus try QFT on Binary 1000 (Decimal 8) in IBM QISKIT Quantum simulator https://qiskit.org/
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  Quantum Phase Estimation(QPE) No of bits to meet accuracy • Quantum phase estimation algorithm (also referred to as quantum eigenvalue estimation algorithm), is a quantum algorithm to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. 𝑈 𝜑 = 𝑒2𝜋𝑖𝜃 𝜑 where U is a unitary operator and 𝑒2𝜋𝑖𝜃 is its eigen value and | 𝜑 is its eigen vector After applying Hadamard gates on counting qubits, 𝜓1 = 1 2 𝑛 2 (|0 + |1 )⊗𝑛|𝜓 After applying Controlled Unitary Operator U for n times, 𝜓2 = 1 2 𝑛 2 |0 + 𝑒2𝜋𝑖𝜃20 |1 ⊗ |0 + 𝑒2𝜋𝑖𝜃21 |1 … … . .⊗ |0 + 𝑒2𝜋𝑖𝜃2𝑛−1 |1 ⊗ |𝜓 𝜓1 𝜓2
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  Quantum Phase Estimation Afterapplying Controlled Unitary Operator U n times, 𝜓2 = 1 2 𝑛 2 |0 + 𝑒2𝜋𝑖𝜃20 |1 ⊗ |0 + 𝑒2𝜋𝑖𝜃21 |1 … … . .⊗ |0 + 𝑒2𝜋𝑖𝜃2𝑛−1 |1 ⊗ |𝜓 = 1 2 𝑛 2 𝑘=0 2𝑛−1 𝑒2𝜋𝑖𝜃𝑘|𝑘 ⊗ |𝜓 This looks similar to QFT equation seen earlier. After applying inverse QFT and making measurement, we will get the output as 2𝑛𝜃 with high probability We will run QPE with T gate given by 1 0 0 𝑒 𝑖𝜋 4 operating a qubit with state |1 getting T |1 = 𝑒2𝜋𝑖𝜃|𝜓 , resulting in 𝜃 = 1/8. We will use 3 counting Qbits and divide the number that we get after measurement by 23, we will get 𝜃. https://qiskit.org/
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  Factoring Problem  Letus try to factor 15 (N) using Shor’s algorithm  We will assume ‘a’ as 7 and find out period ‘r’ (74 = 1 mod(15))  We will find the factors from GCD(𝑥𝑟/2 + 1, 𝑁) 𝑎𝑛𝑑 GCD(𝑥𝑟/2 − 1, 𝑁) https://qiskit.org/
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  Factoring in QC– Current Status  Till 2012, the largest number that was factored by Shor’s algorithm was 21 with 10 qubits. Now different algorithms are being tried with more qubits.  A start-up company Zapata has worked with IBM to develop a new way and factored 1,099,551,473,989 to get prime factors 1,048,589 and 1,048,601 (quote from “New Scientist”)  To break RSA 2048, we would need thousands of logical qubits, but millions of physical qubits (quote from “The Quantum Daily”)  IBM is building a quantum processor with more than 1,000 physical qubits — and somewhere between 10 and 50 logical qubits — by the end of 2023 (quote from “Techcrunch”)
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  References  Basics &Shor algorithm explained https://arxiv.org/abs/quant-ph/9508027 https://www.youtube.com/playlist?list=PL74Rel4IAsETUwZS_Se_P-fSEyEVQwni7 https://en.wikipedia.org/wiki/Shor%27s_algorithm  Different codes https://qiskit.org/textbook/ch-algorithms/shor.html https://github.com/quantumlib/Cirq/blob/master/examples/shor.py  Text Book Quantum Computation and Quantum Information by Michael A Nielsen and Isaax L. Chuang
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  What Next? YOU • Youcan start getting into the depth of quantum computing. • You can go through the references and other videos/ documents and let me know if I need to bring out a video on any specific topic I • I will continue to learn and come out with videos which will be useful for your learning
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  THANK YOU