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Christof Zalka

Publications and source records attributed to Christof Zalka.

18 recordsLinked to original sources
arXiv2006

Shor's algorithm with fewer (pure) qubits

In this note we consider optimised circuits for implementing Shor's quantum factoring algorithm. First I give a circuit for which none 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.

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arXiv2005

Thresholds for Linear Optics Quantum Computing with Photon Loss at the Detectors

We calculate the error threshold for the linear optics quantum computing proposal by Knill, Laflamme and Milburn [Nature 409, pp. 46--52 (2001)] under an error model where photon detectors have efficiency <100% but all other components -- such as single photon sources, beam splitters and phase shifters -- are perfect and introduce no errors. We make use of the fact that the error model induced by the lossy hardware is that of an erasure channel, i.e., the error locations are always known. Using a method based on a Markov chain description of the error correction procedure, our calculations show that, with the 7 qubit CSS quantum code, the gate error threshold for fault tolerant quantum computation is bounded below by a value between 1.78% and 11.5% depending on the construction of the entangling gates.

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arXiv2004

Implementing high dimensional unitary representations of SU(2) on a Quantum Computer

In this note we consider a system with a large angular momentum l whose state we can store using some log_2(l) qubits. The problem then is how to carry out spatial rotations of the system in this representation. In other words we are looking at a unitary representation of SU(2) with dimension 2l+1 and want to implement these transformations with resources polynomial in log(l). We only give a sketch of our solution which involves ``storing'' discretised spherical harmonic functions Y_{l,m}(Theta,phi) in a quantum register. Also there are some technical gaps in the construction, but they are based on plausible assumptions. Our approach is rather cumbersome and we hope somebody will find a nicer solution. For a nice, elementary explanation of what we are trying to do (not involving physics or representation theory) see section 4.6.2.

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arXiv2004

Optimized quantum implementation of elliptic curve arithmetic over binary fields

Shor's quantum algorithm for discrete logarithms applied to elliptic curve groups forms the basis of a "quantum attack" of elliptic curve cryptosystems. To implement this algorithm on a quantum computer requires the efficient implementation of the elliptic curve group operation. Such an implementation requires we be able to compute inverses in the underlying field. In [PZ03], Proos and Zalka show how to implement the extended Euclidean algorithm to compute inverses in the prime field GF(p). They employ a number of optimizations to achieve a running time of O(n^2), and a space-requirement of O(n) qubits (there are some trade-offs that they make, sacrificing a few extra qubits to reduce running-time). In practice, elliptic curve cryptosystems often use curves over the binary field GF(2^m). In this paper, we show how to implement the extended Euclidean algorithm for polynomials to compute inverses in GF(2^m). Working under the assumption that qubits will be an `expensive' resource in realistic implementations, we optimize specifically to reduce the qubit space requirement, while keeping the running-time polynomial. Our implementation here differs from that in [PZ03] for GF(p), and we are able to take advantage of some properties of the binary field GF(2^m). We also optimize the overall qubit space requirement for computing the group operation for elliptic curves over GF(2^m) by decomposing the group operation to make it "piecewise reversible" (similar to what is done in [PZ03] for curves over GF(p)).

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arXiv2004

Shor's discrete logarithm quantum algorithm for elliptic curves

We show in some detail how to implement Shor's efficient 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 security-wise 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($2^n$) or other finite fields. The main technical difficulty 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 difficulty encountered is the ``quantum halting problem''.

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arXiv2003

Exact quantum Fourier transforms and discrete logarithm algorithms

We show how the quantum fast Fourier transform (QFFT) can be made exact for arbitrary orders (first for large primes). For most quantum algorithms only the quantum Fourier transform of order $2^n$ is needed, and this can be done exactly. Kitaev \cite{kitaev} showed how to approximate the Fourier transform for any order. Here we show how his construction can be made exact by using the technique known as ``amplitude amplification''. Although unlikely to be of any practical use, this construction e.g. allows to make Shor's discrete logarithm quantum algorithm exact. Thus we have the first example of an exact non black box fast quantum algorithm, thereby giving more evidence that ``quantum'' need not be probabilistic. We also show that in a certain sense the family of circuits for the exact QFFT is uniform. Namely the parameters of the gates can be calculated efficiently.

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arXiv2002

Comment on "Quantum optimization for combinatorial searches"

This is a comment on a recent publication claiming to have found a ``quantum optimization'' algorithm which outperforms known algorithms for minimizing some ``cost function''. Unfortunately, this algorithm is no better than choosing a state at random and checking whether it has low cost.

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arXiv2001

Quantum operations that cannot be implemented using a small mixed environment

To implement any quantum operation (a.k.a. ``superoperator'' or ``CP map'') on a d-dimensional quantum system, it is enough to apply a suitable overall unitary transformation to the system and a d^2-dimensional environment which is initialized in a fixed pure state. It has been suggested that a d-dimensional environment might be enough if we could initialize the environment in a mixed state of our choosing. In this note we show with elementary means that certain explicit quantum operations cannot be realized in this way. Our counterexamples map some pure states to pure states, giving strong and easily manageable conditions on the overall unitary transformation. Everything works in the more general setting of quantum operations from d-dimensional to d'-dimensional spaces, so we place our counterexamples within this more general framework.

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arXiv2001

Comment on "Stable Quantum Computation of Unstable Classical Chaos"

This is a 1-page comment on a wrong paper that recently appeared in PRL (Phys. Rev. Lett. 86 (23), 5393 (2001), also quant-ph/0101004). The authors claim to have shown that using a quantum computer gives an "exponential advantage" for simulating and studying classical chaotic systems.

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arXiv2000

On the Entangling Power of Quantum Evolutions

We analyze the entangling capabilities of unitary transformations $U$ acting on a bipartite $d_1\times d_2$-dimensional quantum system. To this aim we introduce an entangling power measure $e(U)$ given by the mean linear entropy produced acting with $U$ on a given distribution of pure product states. This measure admits a natural interpretation in terms of quantum operations. For a uniform distribution explicit analytical results are obtained using group-theoretic arguments. The behaviour of the features of $e(U)$ as the subsystem dimensions $d_1$ and $d_2$ are varied is studied both analytically and numerically. The two-qubit case $d_1=d_2=2$ is argued to be peculiar.

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arXiv1999

Grover's quantum searching algorithm is optimal

I improve the tight bound on quantum searching by Boyer et al. (quant-ph/9605034) to a matching bound, thus showing that for any probability of success Grovers quantum searching algorithm is optimal. E.g. for near certain success we have to query the oracle pi/4 sqrt{N} times, where N is the size of the search space. I also show that unfortunately quantum searching cannot be parallelized better than by assigning different parts of the search space to independent quantum computers. Earlier results left open the possibility of a more efficient parallelization.

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arXiv1999

A Grover-based quantum search of optimal order for an unknown number of marked elements

We want to find a marked element out of a black box containing N elements. When the number of marked elements is known this can be done elegantly with Grover's algorithm, a variant of which even gives a correct result with certainty. On the other hand, when the number of marked elements is not known the problem becomes more difficult. For every prescribed success probability I give an algorithm consisting of several runs of Grover's algorithm that matches a recent bound by Buhrman and de Wolf on the order of the number of queries to the black box. The improvement in the order over a previously known algorithm is small and the number of queries can clearly still be reduced by a constant factor.

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arXiv1998

An Introduction to Quantum Computers

This is a short introduction to quantum computers, quantum algorithms and quantum error correcting codes. Familiarity with the principles of quantum theory is assumed. Emphasis is put on a concise presentation of the principles avoiding lengthy discussions.

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arXiv1998

Fast versions of Shor's quantum factoring algorithm

We present fast and highly parallelized versions of Shor's algorithm. With a sizable quantum computer it would then be possible to factor numbers with millions of digits. The main algorithm presented here uses FFT-based fast integer multiplication. The quick reader can just read the introduction and the ``Results'' section.

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arXiv1997

Threshold Estimate for Fault Tolerant Quantum Computation

I make a rough estimate of the accuracy threshold for fault tolerant quantum computing with concatenated codes. First I consider only gate errors and use the depolarizing channel error model. I will follow P.Shor (quant-ph/9505011) for fault tolerant error correction (FTEC) and the fault tolerant implementation of elementary operations on states encoded by the 7-qubit code. A simple computer simulation suggests a threshold for gate errors of the order ε\approx 10^{-3} or better. I also give a simple argument that the threshold for memory errors is about 10 times smaller, thus ε\approx 10^{-4}.

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arXiv1996

Efficient Simulation of Quantum Systems by Quantum Computers

We show that the time evolution of the wave function of a quantum mechanical many particle system can be implemented very efficiently on a quantum computer. The computational cost of such a simulation is comparable to the cost of a conventional simulation of the corresponding classical system. We then sketch how results of interest, like the energy spectrum of a system, can be obtained. We also indicate that ultimately the simulation of quantum field theory might be possible on large quantum computers. We want to demonstrate that in principle various interesting things can be done. Actual applications will have to be worked out in detail also depending on what kind of quantum computer may be available one day...

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