We present a novel and efficient in terms of circuit depth design for Shor's quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the quantum Fourier transform (QFT) Draper's adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively architected into a generic modular quantum multiplier which is the fundamental block for modular exponentiation circuit, the most computational intensive part of Shor's algorithm. The proposed modular exponentiation circuit has a depth of about $2000n^{2}$ and requires $9n+2$ qubits, where $n$ is the number of bits of the classical number to be factored. The total quantum cost of the proposed design is $1600n^{3}$ . The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.We present a novel and efficient in terms of circuit depth design for Shor's quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the quantum Fourier transform (QFT) Draper's adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively architected into a generic modular quantum multiplier which is the fundamental block for modular exponentiation circuit, the most computational intensive part of Shor's algorithm. The proposed modular exponentiation circuit has a depth of about $2000n^{2}$ and requires $9n+2$ qubits, where $n$ is the number of bits of the classical number to be factored. The total quantum cost of the proposed design is $1600n^{3}$. The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.

arXiv:1207.0511Jul 2, 201213 citations

Fast quantum modular exponentiation architecture for Shor's factoring alogrithm

Archimedes Pavlidis, Dimitris Gizopoulos

We present a novel and efficient, in terms of circuit depth, design for Shor's quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the Quantum Fourier transform (QFT) Draper's adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively architected into a quantum modular multiplier which is the fundamental block for the modular exponentiation circuit, the most computational intensive part of Shor's algorithm. The proposed modular exponentiation circuit has a depth of about $2000n^2$ and requires $9n+2$ qubits, where $n$ is the number of bits of the classic number to be factored. The total quantum cost of the proposed design is $1600n^3$ . The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.We present a novel and efficient, in terms of circuit depth, design for Shor's quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the Quantum Fourier transform (QFT) Draper's adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively architected into a quantum modular multiplier which is the fundamental block for the modular exponentiation circuit, the most computational intensive part of Shor's algorithm. The proposed modular exponentiation circuit has a depth of about $2000n^2$ and requires $9n+2$ qubits, where $n$ is the number of bits of the classic number to be factored. The total quantum cost of the proposed design is $1600n^3$. The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.

Quantum Information and ComputationMay 1, 201438 citations

Minimum Depth Quantum Modular Addition Through Carry-Save Architecture

Siyi Wang, Eugene Lim, Xiufan Li +2 more

Shor's factorization algorithm, as one of the most significant achievements in quantum computing, exhibits an exponential speedup compared to the corresponding classical algorithm. In Shor's factorization algorithm, modular exponentiation is one of the most computationally intensive components, which relies on the modular addition building block. This work aims to explore novel designs for enhancing the efficiency of quantum modular addition. In particular, we introduce a novel quantum modular addition framework based on carry-save architecture, which facilitates the conversion of multiple 2-addend quantum operations within modular addition into a single 3-addend operation, thereby reducing the computational depth. Compared to the most efficient existing quantum modular addition, our design has achieved an impressive result - a reduction in Toffoli Depth by up to 33.33%, while maintaining comparable Toffoli Count and Qubit Count. This research underscores the potential of carry-save architecture as a promising technique for accelerating quantum modular arithmetic as well as advancing the development of quantum computing in general.Shor's factorization algorithm, as one of the most significant achievements in quantum computing, exhibits an exponential speedup compared to the corresponding classical algorithm. In Shor's factorization algorithm, modular exponentiation is one of the most computationally intensive components, which relies on the modular addition building block. This work aims to explore novel designs for enhancing the efficiency of quantum modular addition. In particular, we introduce a novel quantum modular addition framework based on carry-save architecture, which facilitates the conversion of multiple 2-addend quantum operations within modular addition into a single 3-addend operation, thereby reducing the computational depth. Compared to the most efficient existing quantum modular addition, our design has achieved an impressive result - a reduction in Toffoli Depth by up to 33.33%, while maintaining comparable Toffoli Count and Qubit Count. This research underscores the potential of carry-save architecture as a promising technique for accelerating quantum modular arithmetic as well as advancing the development of quantum computing in general.

IFIP/IEEE International Conference on Very Large Scale IntegrationOct 6, 20245 citations

A Key Component of the Shor's Algorithm and Its Possible Quantum Implementations

Jiří Tomčala

A fundamental problem in factorization by Shor's algorithm [1] is the implementation of the modular exponentiation function. The period of this function is important for determining the individual factors, so its accurate implementation is for successful factorization critical.The imprecision of current quantum computers requires that any quantum circuits should be as simple as possible. Both in terms of the number of qubits used, the number of quantum gates and the depth of the quantum circuit. Therefore, it is necessary to search for an optimal variant for the implementation of the above-mentioned modular exponentiation function.The standard method is to implement individual arithmetic functions [2] which can be combined to achieve the desired modular exponentiation function. This process, however, creates a very complex quantum circuit, so that its measured result (probability distribution) will be so skewed that it will not be possible to reliably obtain the period of sought.Another way is to create a special quantum circuit for one particular modular exponentiation function [3]. Such a circuit is of course simpler than the circuit created by the above universal implementation, but in practice it only makes sense for testing the functionality of the surrounding quantum circuit for period detection.A very interesting possibility for the implementation of arithmetic functions is to use the direct and inverse quantum Fourier transform [4]. For a quantum computer, these transformations are very easy operations, and this interesting feature can then be used to advantage in these various implementations.This contribution will explore these possible approaches and show their advantages and disadvantages through concrete examples.A fundamental problem in factorization by Shor's algorithm [1] is the implementation of the modular exponentiation function. The period of this function is important for determining the individual factors, so its accurate implementation is for successful factorization critical.The imprecision of current quantum computers requires that any quantum circuits should be as simple as possible. Both in terms of the number of qubits used, the number of quantum gates and the depth of the quantum circuit. Therefore, it is necessary to search for an optimal variant for the implementation of the above-mentioned modular exponentiation function.The standard method is to implement individual arithmetic functions [2] which can be combined to achieve the desired modular exponentiation function. This process, however, creates a very complex quantum circuit, so that its measured result (probability distribution) will be so skewed that it will not be possible to reliably obtain the period of sought.Another way is to create a special quantum circuit for one particular modular exponentiation function [3]. Such a circuit is of course simpler than the circuit created by the above universal implementation, but in practice it only makes sense for testing the functionality of the surrounding quantum circuit for period detection.A very interesting possibility for the implementation of arithmetic functions is to use the direct and inverse quantum Fourier transform [4]. For a quantum computer, these transformations are very easy operations, and this interesting feature can then be used to advantage in these various implementations.This contribution will explore these possible approaches and show their advantages and disadvantages through concrete examples.

Dec 17, 20250 citations

Low-depth quantum architectures for factoring

Aram W. Harrow, Paul Pham

Quantum computing is a new field which combines computer science and quantum physics. Its most famous result, Shor's factoring algorithm, would enable us to one day compromise the widely-used RSA cryptosystem if we are able to design efficient quantum architectures. Studying the depth of these architectures would allow us to solve this human problem within a human lifetime. Toward that end, we contribute two hybrid factoring architectures in poly-logarithmic and sub-logarithmic depths, which are both exponential improvements over previous known works. We also present an improved procedure for generating quantum Fourier states useful for quantum compiling. Finally, we invent a new circuit resource called coherence which upper bounds the error-correcting effort needed in a future quantum computer. We use this to characterize a better time-space tradeoff for factoring as well as to provide configurable-depth factoring architectures.Quantum computing is a new field which combines computer science and quantum physics. Its most famous result, Shor's factoring algorithm, would enable us to one day compromise the widely-used RSA cryptosystem if we are able to design efficient quantum architectures. Studying the depth of these architectures would allow us to solve this human problem within a human lifetime. Toward that end, we contribute two hybrid factoring architectures in poly-logarithmic and sub-logarithmic depths, which are both exponential improvements over previous known works. We also present an improved procedure for generating quantum Fourier states useful for quantum compiling. Finally, we invent a new circuit resource called coherence which upper bounds the error-correcting effort needed in a future quantum computer. We use this to characterize a better time-space tradeoff for factoring as well as to provide configurable-depth factoring architectures.

Jan 1, 20130 citations

Faster quantum number factoring via circuit synthesis

Igor L. Markov, Mehdi Saeedi

A major obstacle to implementing Shor's quantum number-factoring algorithm is the large size of modular-exponentiation circuits. We reduce this bottleneck by customizing reversible circuits for modular multiplication to individual runs of Shor's algorithm. Our circuit-synthesis procedure exploits spectral properties of multiplication operators and constructs optimized circuits from the traces of the execution of an appropriate greatest-common-divisor algorithm. Empirically, gate counts are reduced by four to five times, and circuit latency is reduced by larger factors.A major obstacle to implementing Shor's quantum number-factoring algorithm is the large size of modular-exponentiation circuits. We reduce this bottleneck by customizing reversible circuits for modular multiplication to individual runs of Shor's algorithm. Our circuit-synthesis procedure exploits spectral properties of multiplication operators and constructs optimized circuits from the traces of the execution of an appropriate greatest-common-divisor algorithm. Empirically, gate counts are reduced by four to five times, and circuit latency is reduced by larger factors.

arXiv:1301.3210Physical Review AJan 14, 201334 citations

2000n^{2}

and requires

9n+2

qubits, where

n

is the number of bits of the classical number to be factored. The total quantum cost of the proposed design is

1600n^{3}

. The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.

2000n^2

and requires

9n+2

qubits, where

n

is the number of bits of the classic number to be factored. The total quantum cost of the proposed design is

1600n^3

. The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.