Shor's algorithm, developed by Peter Shor in 1994, efficiently finds prime factors of integers and poses a threat to RSA cryptography. The document explains the prerequisites, the quantum phase estimation (QPE) algorithm, and the quantum Fourier transform (QFT) necessary for implementing Shor's algorithm. Current advances in quantum computing are discussed, including successful factorizations and the need for extensive qubits to break RSA-2048 encryption.

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Shors'algorithm simplified.pptx

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  Quantum ComputationSimplifiedKathiresan SPart- 6 Shor’s algorithm, QPE & QFT
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  IntroductionWhat is Shor’salgorithm? This is to find prime factors of an integer and invented by an Americanmathematician 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 manyyears in classical computer and a few seconds in a perfect Quantum Computer)
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  Prerequisite for thisvideoEssential Basic knowledge of Quantum Computing – my previous videos (Part 1 to 5) orequivalent from other videos/ textbooksDesirable 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 roughlyequal 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 GreatestCommon 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 conditionsare not met. Probability of guessing the right x is 1 −12𝑘−1 where k is the number of distinct odd primefactors 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 thatxr 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 itseigen vector In order to measure 𝜃, we need to know about Quantum Fourier Transform (& itsinverse) and Continued Fraction
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  Quantum Fourier Transform(QFT) Similar to FFT/ DFT, transforms between two bases, the computational (Z) basis, andthe Fourier basis8 (decimal) in Computational Basis8 (decimal) in Fourier BasisRotation of qubits – Multiples of82𝑛 x 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =816x 360 𝑑𝑒𝑔𝑟𝑒𝑒𝑠15 (decimal) in Computational Basis15 (decimal) in Fourier Basis
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  Quantum Fourier TransformDiscrete Fourier TransformDFT of a vector of complex numbers, x0,….,xN-1 is given by the following equation wherein yk = y0,…...yN-1𝑦𝑘 ≡1𝑁 𝑗=0𝑁−1𝑥𝑗𝑒2𝜋𝑖𝑗𝑘/𝑁 Quantum Fourier TransformQFT on an orthonormal basis states |0 , … … , |𝑁 − 1 is given by the following equation|𝑗 →1𝑁 𝑘=0𝑁−1𝑒2𝜋𝑖𝑗𝑘/𝑁|𝑘Assuming N = 2n where n is an integer,|𝑗 →12𝑛/2𝑘=02𝑛−1𝑒2𝜋𝑖𝑗𝑘/2𝑛|𝑘
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  Quantum Fourier TransformFromprevious slide,|𝑗 →12𝑛/2𝑘=02𝑛−1𝑒2𝜋𝑖𝑗𝑘/2𝑛|𝑘Normally, 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|12𝑛/2
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  Quantum Fourier TransformFromprevious slide,|𝑗1,……….𝑗𝑛→|0 + 𝑒2𝜋𝑖0.𝑗𝑛|1 |0 + 𝑒2𝜋𝑖0.𝑗𝑛−1𝑗𝑛|1 … … |0 + 𝑒2𝜋𝑖0.𝑗1𝑗2……𝑗𝑛|12𝑛/2𝑤ℎ𝑒𝑟𝑒𝑖𝑛 0. 𝑗1𝑗2 ….. 𝑗𝑛 =𝑗121 +𝑗222 + ⋯ … … +𝑗𝑛2𝑛This can be achieved using three types of gates - Hadamard gate121 11 −1and aRotational gate 𝑅𝑘 =1 00 𝑒2𝜋𝑖2𝑘𝑤ℎ𝑒𝑟𝑒 𝑒2𝜋𝑖2𝑘is 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 TransformLetus try QFT on Binary 1000 (Decimal 8) in IBM QISKIT Quantum simulatorhttps://qiskit.org/
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  Quantum Phase Estimation(QPE)No of bits tomeet accuracy• Quantum phase estimation algorithm (also referred to as quantum eigenvalue estimation algorithm), isa quantum algorithm to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. 𝑈 𝜑 = 𝑒2𝜋𝑖𝜃 𝜑where U is a unitary operator and 𝑒2𝜋𝑖𝜃is its eigen value and | 𝜑 is its eigen vectorAfter applying Hadamard gates on counting qubits, 𝜓1 =12𝑛2(|0 + |1 )⊗𝑛|𝜓After applying Controlled Unitary Operator U for n times, 𝜓2 =12𝑛2|0 + 𝑒2𝜋𝑖𝜃20|1 ⊗ |0 + 𝑒2𝜋𝑖𝜃21|1 … … . .⊗|0 + 𝑒2𝜋𝑖𝜃2𝑛−1|1 ⊗ |𝜓𝜓1 𝜓2
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  Quantum Phase EstimationAfterapplying Controlled Unitary Operator U n times,𝜓2 =12𝑛2|0 + 𝑒2𝜋𝑖𝜃20|1 ⊗ |0 + 𝑒2𝜋𝑖𝜃21|1 … … . .⊗ |0 + 𝑒2𝜋𝑖𝜃2𝑛−1|1 ⊗ |𝜓 =12𝑛2𝑘=02𝑛−1𝑒2𝜋𝑖𝜃𝑘|𝑘 ⊗ |𝜓This looks similar to QFT equation seen earlier. After applying inverse QFT and making measurement,we will get the output as 2𝑛𝜃 with high probabilityWe will run QPE with T gate given by1 00 𝑒𝑖𝜋4operating 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 10qubits. Now different algorithms are being tried with more qubits. A start-up company Zapata has worked with IBM to develop a new way and factored1,099,551,473,989 to get prime factors 1,048,589 and 1,048,601 (quote from “NewScientist”) To break RSA 2048, we would need thousands of logical qubits, but millions ofphysical qubits (quote from “The Quantum Daily”) IBM is building a quantum processor with more than 1,000 physical qubits — andsomewhere between 10 and 50 logical qubits — by the end of 2023 (quote from“Techcrunch”)
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  References Basics &Shor algorithm explainedhttps://arxiv.org/abs/quant-ph/9508027https://www.youtube.com/playlist?list=PL74Rel4IAsETUwZS_Se_P-fSEyEVQwni7https://en.wikipedia.org/wiki/Shor%27s_algorithm Different codeshttps://qiskit.org/textbook/ch-algorithms/shor.htmlhttps://github.com/quantumlib/Cirq/blob/master/examples/shor.py Text BookQuantum 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 letme know if I need to bring out a video on any specific topicI• I will continue to learn and come out with videos which will be useful foryour learning
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  THANK YOU