Quantum chaos in the spectrum of operators used in Shor's algorithm

We provide compelling evidence for the presence of quantum chaos in the unitary part of Shor's factoring algorithm. In particular we analyze the spectrum of this part after proper desymmetrization and show that the fluctuations of the eigenangles as well as the distribution of the eigenvector components follow the CUE ensemble of random matrices, of relevance to quantized chaotic systems that violate time-reversal symmetry. However, as the algorithm tracks the evolution of a single state, it is possible to employ other operators, in particular it is possible that the generic quantum chaos found above becomes of a nongeneric kind such as is found in the quantum cat maps, and in toy models of the quantum bakers map.Comment: Title and paper modified to include interesting additional possibilities. Principal results unaffected. Accepted for publication in Phys. Rev. E as Rapid Com

Appearance and Stability of Anomalously Fluctuating States in Shor's Factoring Algorithm

We analyze quantum computers which perform Shor's factoring algorithm, paying attention to asymptotic properties as the number L of qubits is increased. Using numerical simulations and a general theory of the stabilities of many-body quantum states, we show the following: Anomalously fluctuating states (AFSs), which have anomalously large fluctuations of additive operators, appear in various stages of the computation. For large L, they decohere at anomalously great rates by weak noises that simulate noises in real systems. Decoherence of some of the AFSs is fatal to the results of the computation, whereas decoherence of some of the other AFSs does not have strong influence on the results of the computation. When such a crucial AFS decoheres, the probability of getting the correct computational result is reduced approximately proportional to L^2. The reduction thus becomes anomalously large with increasing L, even when the coupling constant to the noise is rather small. Therefore, quantum computations should be improved in such a way that all AFSs appearing in the algorithms do not decohere at such great rates in the existing noises.Comment: 11 figures. A few discussions were added in verion 2. Version 3 is the SAME as version 2; only errors during the Web-upload were fixed. Version 4 is the publised version, in which several typos are fixed and the reference list is update

Spectral modular arithmetic

In many areas of engineering and applied mathematics, spectral methods provide very powerful tools for solving and analyzing problems. For instance, large to extremely large sizes of numbers can efficiently be multiplied by using discrete Fourier transform and convolution property. Such computations are needed when computing π to millions of digits of precision, factoring and also big prime search projects. When it comes to the utilization of spectral techniques for modular operations in public key cryptosystems two difficulties arise; the first one is the reduction needed after the multiplication step and the second is the cryptographic sizes which are much shorter than the optimal asymptotic crossovers of spectral methods. In this dissertation, a new modular reduction technique is proposed. Moreover, modular multiplication is given based on this reduction. These methods work fully in the frequency domain with some exceptions such as the initial, final and partial transformations steps. Fortunately, the new technique addresses the reduction problem however, because of the extra complexity coming from the overhead of the forward and backward transformation computations, the second goal is not easily achieved when single operations such as modular multiplication or reduction are considered. On the contrary, if operations that need several modular multiplications with respect to the same modulus are considered, this goal is more tractable. An obvious example of such an operation is the modular exponentiation i.e., the computation of c=m[superscript e] mod n where c, m, e, n are large integers. Therefore following the spectral modular multiplication operation a new modular exponentiation method is presented. Since forward and backward transformation calculations do not need to be performed for every multiplication carried during the exponentiation, the asymptotic crossover for modular exponentiation is decreased to cryptographic sizes. The method yields an efficient and highly parallel architecture for hardware implementations of public-key cryptosystems

Constructing KMS states from infinite-dimensional spectral triples

We construct KMS-states from $\mathrm{Li}_1$-summable semifinite spectral triples and show that in several important examples the construction coincides with well-known direct constructions of KMS-states for naturally defined flows. Under further summability assumptions the constructed KMS-state can be computed in terms of Dixmier traces. For closed manifolds, we recover the ordinary Lebesgue integral. For Cuntz-Pimsner algebras with their gauge flow, the construction produces KMS-states from traces on the coefficient algebra and recovers the Laca-Neshveyev correspondence. For a discrete group acting on its Stone-\v{C}ech boundary, we recover the Patterson-Sullivan measures on the Stone-\v{C}ech boundary for a flow defined from the Radon-Nikodym cocycle.Comment: 66 page

A simplification of the Shor quantum factorization algorithm employing a quantum Hadamard transform

The Shor quantum factorization algorithm allows the factorization or large integers in logarithmic squared time whereas classical algorithms require an exponential time increase with the bit length of the number to be factored. The hardware implementation of the Shor algorithm would thus allow the factorization of the very large integers employed by commercial encryption methods. We propose some modifications of the algorithm by employing some simplification to the stage employing the quantum Fourier transform. The quantum Hadamard transform may be used to replace the quantum Fourier transform in certain cases. This would reduce the hardware complexity of implementation since phase rotation gates with only two states of 0 and π would be required

Probing thermality beyond the diagonal

We investigate the off-diagonal sector of eigenstate thermalization using both local and non-local probes in 2-dimensional conformal field theories. A novel analysis of the asymptotics of OPE coefficients via the modular bootstrap is performed to extract the behaviour of the off-diagonal matrix elements. We also probe this sector using semi-classical heavy-light Virasoro blocks. The results demonstrate signatures of thermality and confirms the entropic suppression of the off-diagonal elements as necessitated by the eigenstate thermalization hypothesis.Comment: 27 pages, 2 figure