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Shor's Discrete Logarithm Quantum Algorithm for Elliptic Curves
, 2003
"... We show in some detail how to implement Shor's e#cient quantum algorithm for discrete logarithms for the particular case of elliptic curve groups. It turns out that for this problem a smaller quantum computer can solve problems further beyond current computing than for integer factorisation. A 1 ..."
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We show in some detail how to implement Shor's e#cient quantum algorithm for discrete logarithms for the particular case of elliptic curve groups. It turns out that for this problem a smaller quantum computer can solve problems further beyond current computing than for integer factorisation. A 160 bit elliptic curve cryptographic key could be broken on a quantum computer using around 1000 qubits while factoring the securitywise equivalent 1024 bit RSA modulus would require about 2000 qubits. In this paper we only consider elliptic curves over GF(p) and not yet the equally important ones over GF(2 ) or other finite fields. The main technical di#culty is to implement Euclid's gcd algorithm to compute multiplicative inverses modulo p. As the runtime of Euclid's algorithm depends on the input, one di#culty encountered is the "quantum halting problem". On an (even) more theoretical note we also point out that there are quantum circuits which make the discrete logarithm algorithm exact.
Shor’s algorithm on a nearestneighbor machine
 Asian conference on Quantum Information Science
, 2007
"... We give a new “nested adds ” circuit for implementing Shor’s algorithm in linear width and quadratic depth on a nearestneighbor machine. Our circuit combines Draper’s transform adder with approximation ideas of Zalka. The transform adder requires small controlled rotations. We also give another ver ..."
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Cited by 9 (1 self)
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We give a new “nested adds ” circuit for implementing Shor’s algorithm in linear width and quadratic depth on a nearestneighbor machine. Our circuit combines Draper’s transform adder with approximation ideas of Zalka. The transform adder requires small controlled rotations. We also give another version, with slightly larger depth, using only reversible classical gates. We do not know which version will ultimately be cheaper to implement. 1
Shor’s algorithm with fewer (pure) qubits
, 2006
"... In this note we consider optimised circuits for implementing Shor’s quantum factoring algorithm. First I give a circuit for which non of the about 2n qubits need to be initialised (though we still have to make the usual 2n measurements later on). Then I show how the modular additions in the algorith ..."
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Cited by 1 (0 self)
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In this note we consider optimised circuits for implementing Shor’s quantum factoring algorithm. First I give a circuit for which non of the about 2n qubits need to be initialised (though we still have to make the usual 2n measurements later on). Then I show how the modular additions in the algorithm can be carried out with a superposition of an arithmetic sequence. This makes parallelisation of Shor’s algorithm easier. Finally I show how one can factor with only about 1.5n qubits, and maybe even fewer. 1
and
, 2007
"... We evaluate the performance of quantum arithmetic algorithms run on a distributed quantum computer (a quantum multicomputer). We vary the node capacity and I/O capabilities, and the network topology. The tradeoff of choosing between gates executed remotely, through “teleported gates ” on entangled p ..."
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We evaluate the performance of quantum arithmetic algorithms run on a distributed quantum computer (a quantum multicomputer). We vary the node capacity and I/O capabilities, and the network topology. The tradeoff of choosing between gates executed remotely, through “teleported gates ” on entangled pairs of qubits (telegate), versus exchanging the relevant qubits via quantum teleportation, then executing the algorithm using local gates (teledata), is examined. We show that the teledata approach performs better, and that carryripple adders perform well when the teleportation block is decomposed so that the key quantum operations can be parallelized. A node size of only a few logical qubits performs adequately provided that the nodes have two transceiver qubits. A linear network topology performs acceptably for a broad range of system sizes and performance parameters. We therefore recommend pursuing small, highI/O bandwidth nodes and a simple network. Such a machine will run Shor’s algorithm for factoring large numbers efficiently.
Topology in Physics
, 2006
"... 1 Lie Algebras: a crash course................ 1 Cyril Stark (under the supervision of Urs Wenger) 2 Goldstone bosons and the Higgs mechanism.... 51 Stefan Pfenninger (under the supervision of Stefan Fredenhagen) ..."
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1 Lie Algebras: a crash course................ 1 Cyril Stark (under the supervision of Urs Wenger) 2 Goldstone bosons and the Higgs mechanism.... 51 Stefan Pfenninger (under the supervision of Stefan Fredenhagen)
Simulating Special but Natural Quantum Circuits
"... We identify a subclass of BQP that captures certain structural commonalities among many quantum algorithms including Shor’s algorithms. This class does not contain all of BQP (e.g. Grover’s algorithm does not fall into this class). Our main result is that any algorithm in this class that measures a ..."
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We identify a subclass of BQP that captures certain structural commonalities among many quantum algorithms including Shor’s algorithms. This class does not contain all of BQP (e.g. Grover’s algorithm does not fall into this class). Our main result is that any algorithm in this class that measures at most O(log n) qubits can be simulated by classical randomized polynomial time algorithms. This does not dequantize Shor’s algorithm (as the latter measures n qubits) but our work also highlights a new potentially hard function for cryptographic applications. Our main technical contribution is (to the best of our knowledge) a new exact characterization of certain sums of Fouriertype coefficients (with exponentially many summands). One of the key problems in complexity theory is to determine the power of the complexity class BQP. Recall that this is the set of languages accepted by uniform polynomial size quantum circuits with bounded twosided error. It is essentially the quantum version of the complexity class BPP. Just as BPP corresponds to what is feasible on a classical computer with randomness, BQP corresponds
Entanglement and its Role in Shor’s Algorithm
, 2005
"... Entanglement has been termed a critical resource for quantum information processing and is thought to be the reason that certain quantum algorithms, such as Shor’s factoring algorithm, can achieve exponentially better performance than their classical counterparts. The nature of this resource is stil ..."
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Entanglement has been termed a critical resource for quantum information processing and is thought to be the reason that certain quantum algorithms, such as Shor’s factoring algorithm, can achieve exponentially better performance than their classical counterparts. The nature of this resource is still not fully understood: here we use numerical simulation to investigate how entanglement between register qubits varies as Shor’s algorithm is run on a quantum computer. The shifting patterns in the entanglement are found to relate to the choice of basis for the quantum Fourier transform. Key words: Quantum computing 1
c ○ Rinton Press A 2D NEARESTNEIGHBOR QUANTUM ARCHITECTURE FOR FACTORING IN POLYLOGARITHMIC DEPTH
, 2012
"... We present a 2D nearestneighbor quantum architecture for Shor’s algorithm to factor an nbit number in O(log 3 n) depth. Our implementation uses parallel phase estimation, constantdepth fanout and teleportation, and constantdepth carrysave modular addition. We derive upper bounds on the circuit ..."
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We present a 2D nearestneighbor quantum architecture for Shor’s algorithm to factor an nbit number in O(log 3 n) depth. Our implementation uses parallel phase estimation, constantdepth fanout and teleportation, and constantdepth carrysave modular addition. We derive upper bounds on the circuit resources of our architecture under a new 2D model which allows a classical controller and parallel, communicating modules. We provide a comparison to all previous nearestneighbor factoring implementations. Our circuit results in an exponential improvement in nearestneighbor circuit depth at the cost of a polynomial increase in circuit size and width.