6 research outputs found
Improved techniques for preparing eigenstates of fermionic Hamiltonians
Modeling low energy eigenstates of fermionic systems can provide insight into chemical reactions and material properties and is one of the most anticipated applications of quantum computing. We present three techniques for reducing the cost of preparing fermionic Hamiltonian eigenstates using phase estimation. First, we report a polylogarithmic-depth quantum algorithm for antisymmetrizing the initial states required for simulation of fermions in first quantization. This is an exponential improvement over the previous state-of-the-art. Next, we show how to reduce the overhead due to repeated state preparation in phase estimation when the goal is to prepare the ground state to high precision and one has knowledge of an upper bound on the ground state energy that is less than the excited state energy (often the case in quantum chemistry). Finally, we explain how one can perform the time evolution necessary for the phase estimation based preparation of Hamiltonian eigenstates with exactly zero error by using the recently introduced qubitization procedure
Random Routing and Concentration in Quantum Switching Networks
Flexible distribution of data in the form of quantum bits or qubits
among spatially separated entities is an essential component of
envisioned scalable quantum computing architectures. Accordingly, we
consider the problem of dynamically permuting groups of quantum bits,
i.e., qubit packets, using networks of reconfigurable quantum
switches.
We demonstrate and then explore the equivalence between the quantum
process of creation of packet superpositions and the process of
randomly routing packets in the corresponding classical network. In
particular, we consider an n × n Baseline network for which we
explicitly relate the pairwise input-output routing probabilities in
the classical random routing scenario to the probability amplitudes of
the individual packet patterns superposed in the quantum output state.
We then analyze the effect of using quantum random routing on a
classically non-blocking configuration like the Benes network. We
prove that for an n × n quantum Benes network, any input
packet assignment with no output contention is probabilistically
self-routable. In particular, we prove that with random routing on the
first (log n-1) stages and bit controlled self-routing on the last
log n stages of a quantum Benes network, the output packet
pattern corresponding to routing with no blocking is always present in
the output quantum state with a non-zero probability. We give a lower
bound on the probability of observing such patterns on measurement at
the output and identify a class of 2n-1 permutation patterns for
which this bound is equal to 1, i.e., for all the permutation
patterns in this class the following is true: in every pattern
in the quantum output assignment all the valid input packets are
present at their correct output addresses.
In the second part of this thesis we give the complete design of
quantum sparse crossbar concentrators. Sparse crossbar concentrators
are rectangular grids of simple 2 × 2 switches or crosspoints,
with the switches arranged such that any k inputs can be connected
to some k outputs. We give the design of the quantum crosspoints for
such concentrators and devise a self-routing method to concentrate
quantum packets. Our main result is a rigorous proof that certain
crossbar structures, namely, the fat-slim and banded quantum crossbars
allow, without blocking, the realization of all concentration patterns
with self-routing.
In the last part we consider the scenario in which quantum packets are
queued at the inputs to an n × n quantum non-blocking
switch. We assume that each packet is a superposition of m classical
packets. Under the assumption of uniform traffic, i.e., any output is
equally likely to be accessed by a packet at an input we find the
minimum value of m such that the output quantum state contains at
least one packet pattern in which no two packets contend for the same
output. Our calculations show that for m=9 the probability of a
non-contending output pattern occurring in the quantum output is
greater than 0.99 for all n up to 64
Quantum Algorithmic Techniques for Fault-Tolerant Quantum Computers
Quantum computers have the potential to push the limits of computation in areas such as quantum chemistry, cryptography, optimization, and machine learning. Even though many quantum algorithms show asymptotic improvement compared to classical ones, the overhead of running quantum computers limits when quantum computing becomes useful. Thus, by optimizing components of quantum algorithms, we can bring the regime of quantum advantage closer. My work focuses on developing efficient subroutines for quantum computation. I focus specifically on algorithms for scalable, fault-tolerant quantum computers. While it is possible that even noisy quantum computers can outperform classical ones for specific tasks, high-depth and therefore fault-tolerance is likely required for most applications. In this thesis, I introduce three sets of techniques that can be used by themselves or as subroutines in other algorithms.
The first components are coherent versions of classical sort and shuffle. We require that a quantum shuffle prepares a uniform superposition over all permutations of a sequence. The quantum sort is used within the shuffle and as well as in the next algorithm in this thesis. The quantum shuffle is an essential part of state preparation for quantum chemistry computation in first quantization.
Second, I review the progress of Hamiltonian simulations and give a new algorithm for simulating time-dependent Hamiltonians. This algorithm scales polylogarithmic in the inverse error, and the query complexity does not depend on the derivatives of the Hamiltonian. A time-dependent Hamiltonian simulation was recently used for interaction picture simulation with applications to quantum chemistry.
Next, I present a fully quantum Boltzmann machine. I show that our algorithm can train on quantum data and learn a classical description of quantum states. This type of machine learning can be used for tomography, Hamiltonian learning, and approximate quantum cloning
Quantum Cost Models for Cryptanalysis of Isogenies
Isogeny-based cryptography uses keys large enough to resist a far-future attack from
Tani’s algorithm, a quantum random walk on Johnson graphs. The key size is based on an
analysis in the query model. Queries do not reflect the full cost of an algorithm, and this
thesis considers other cost models. These models fit in a memory peripheral framework,
which focuses on the classical control costs of a quantum computer. Rather than queries,
we use the costs of individual gates, error correction, and latency. Primarily, these costs
make quantum memory access expensive and thus Tani’s memory-intensive algorithm is
no longer the best attack against isogeny-based cryptography. A classical algorithm due to
van Oorschot and Wiener can be faster and cheaper, depending on the model used and the
availability of time and hardware. This means that isogeny-based cryptography is more
secure than previously thought