# Constant-optimized quantum circuits for modular multiplication and exponentiation

@article{Markov2012ConstantoptimizedQC, title={Constant-optimized quantum circuits for modular multiplication and exponentiation}, author={Igor L. Markov and Mehdi Saeedi}, journal={Quantum Inf. Comput.}, year={2012}, volume={12}, pages={361-394} }

Reversible circuits for modular multiplication Cx%M with x < M arise as components of modular exponentiation in Shor's quantum number-factoring algorithm. However, existing generic constructions focus on asymptotic gate count and circuit depth rather than actual values, producing fairly large circuits not optimized for specific C and M values. In this work, we develop such optimizations in a bottom-up fashion, starting with most convenient C values. When zero-initialized ancilla registers are…

## 86 Citations

### Improved reversible and quantum circuits for Karatsuba-based integer multiplication

- Computer ScienceTQC
- 2017

A reversible circuit for integer multiplication that is inspired by Karatsuba's recursive method is presented, with the main improvement over circuits that have been previously reported in the literature is an asymptotic reduction of the amount of space required.

### Optimizing Quantum Circuits for Modular Exponentiation

- Computer Science2019 32nd International Conference on VLSI Design and 2019 18th International Conference on Embedded Systems (VLSID)
- 2019

This paper shows the design of quantum circuit to perform modular exponentiation functions using two different approaches using synthesis tools and then applies two different Quantum Error Correction techniques.

### CNOT-count optimized quantum circuit of the Shor's algorithm

- Computer Science
- 2021

Improved quantum circuit for modular exponentiation of a constant, which is the most expensive operation in Shor’s algorithm for integer factorization, is presented and the lower bound of CNOT numbers needed to implement Shor's algorithm is discussed.

### Reversible logic synthesis of k-input, m-output lookup tables

- Computer Science2013 Design, Automation & Test in Europe Conference & Exhibition (DATE)
- 2013

The proposed LUT synthesis has a significant impact on reducing the size of modular exponentiation circuits for Shor's quantum factoring algorithm, oracle circuits in quantum walk on sparse graphs, and the well-known MCNC benchmarks.

### Improved quantum ripple-carry addition circuit

- Physics, Computer ScienceScience China Information Sciences
- 2015

This work presents an improved linear-depth ripple-carry quantum addition circuit, which is an elementary circuit used for quantum computations, and its modular-multiplication circuits are simpler than previous constructions, and may be used as primitive circuits for quantum computation.

### Faster Quantum Number Factoring via Circuit Synthesis

- Computer ScienceArXiv
- 2013

A circuit-synthesis procedure exploits spectral properties of multiplication operators and constructs optimized circuits from the traces of the execution of an appropriate GCD algorithm, reducing gate counts and circuit latency by up to 4-5 times.

### Simulation of Modular Exponentiation Circuit for Shor's Algorithm in Qiskit

- Computer Science2020 14th International Conference on Telecommunication Systems, Services, and Applications (TSSA
- 2020

This paper discusses and demonstrates the construction of a quantum modular exponentiation circuit in the Qiskit simulator for use in Shor's Algorithm for integer factorization problem (IFP), which…

### Quantum Circuit Design of Toom 3-Way Multiplication

- Computer ScienceApplied Sciences
- 2021

The corresponding quantum circuit for Toom-3 multiplication is designed and adopted the sequence proposed by Bodrato to yield a lower number of operations, especially in terms of nontrivial division, which is reduced to only one exact division by 3 circuit per iteration.

### Low-depth quantum architectures for factoring

- Computer Science
- 2013

A new circuit resource called coherence is invented which upper bounds the error-correcting effort needed in a future quantum computer and is used to characterize a better time-space tradeoff for factoring as well as to provide configurable-depth factoring architectures.

### Simulation of Quantum Circuits via Stabilizer Frames

- Computer ScienceIEEE Transactions on Computers
- 2015

This work develops new data structures and algorithms that facilitate parallel simulation of realistic circuits enriched with quantum error-correcting codes and fault-tolerant procedures, and demonstrates that Quipu can be parallelized with a nontrivial computational speedup.

## References

SHOWING 1-10 OF 28 REFERENCES

### Faster Quantum Number Factoring via Circuit Synthesis

- Computer ScienceArXiv
- 2013

A circuit-synthesis procedure exploits spectral properties of multiplication operators and constructs optimized circuits from the traces of the execution of an appropriate GCD algorithm, reducing gate counts and circuit latency by up to 4-5 times.

### Architecture of a quantum multicomputer optimized for Shor's factoring algorithm

- Computer Science, Physics
- 2006

A number of optimizations for the modular exponentiation of an n-bit number are introduced, including one that reduces the latency, or circuit depth, and two forms of carry-ripple adder found to be the fastest for a wide range of performance parameters.

### Fast quantum modular exponentiation

- Computer Science
- 2005

It is found that to exponentia te an n-bit number, for storage space 100n, the authors can execute modular exponentiation two hundred to seven hundred times faster than optimized versions of the basic algorithms, depending on architecture, for n = 128.

### Quantum networks for elementary arithmetic operations.

- Computer SciencePhysical review. A, Atomic, molecular, and optical physics
- 1996

This work provides an explicit construction of quantum networks effecting basic arithmetic operations: from addition to modular exponentiation, and shows that the auxiliary memory required to perform this operation in a reversible way grows linearly with the size of the number to be factorized.

### Elementary gates for quantum computation.

- MathematicsPhysical review. A, Atomic, molecular, and optical physics
- 1995

U(2) gates are derived, which derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number of unitary operations on arbitrarily many bits.

### Implementation of Shor's algorithm on a linear nearest neighbour qubit array

- Computer ScienceQuantum Inf. Comput.
- 2004

A circuit implementing Shor's factorisation algorithm designed for such a linear nearest neighbour architecture with interaction restrictions, which is identical to leading order to that possible using an architecture that can interact arbitrary pairs of qubits.

### On the CNOT-cost of TOFFOLI gates

- PhysicsQuantum Inf. Comput.
- 2009

It is proved that the n-qubit analogue of the TOFFOLI requires at least 2n CNOT gates, and a complete classification of three-qu bit diagonaloperators by their CNOT-cost, which holds even if ancilla qubits are available.

### Circuit for Shor's algorithm using 2n+3 qubits

- Computer Science, MathematicsQuantum Inf. Comput.
- 2003

A circuit which uses 2n + 3 qubits and 0(n3lg(n)) elementary quantum gates in a depth of 0( n3) to implement the factorization algorithm using Shor's algorithm on a quantum computer.

### BDD-based synthesis of reversible logic for large functions

- Computer Science2009 46th ACM/IEEE Design Automation Conference
- 2009

This paper presents a technique to derive reversible circuits for a function given by a binary decision diagram (BDD), and shows better results and a significantly better scalability in comparison to previous synthesis approaches.

### Quantum addition circuits and unbounded fan-out

- Computer Science, MathematicsQuantum Inf. Comput.
- 2010

These circuits are applied to constructing efficient quantum circuits for Shor's discrete logarithm algorithm and yield efficient circuits with depth O(log n) and with depthO(log* n), respectively.