First some background. One of these is as a quantum walk across a surface—the way a quantum particle would move randomly from one point to another. Clearly, this process is a kind of search of two-dimensional space. But because a quantum particle can explore many paths at the same time, it is much faster than a classical search.

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The nature of the surface has an important influence on the search. For example, one type of surface consists of a square grid where the quantum particle has four possible moves at each vertex. But there are many other possible grids; a triangular one, for example, where the quantum particle has three choices at each vertex.

The team focused on simulating the way a Grover search works for electrons exploring triangular and square grids, but they also included other physically realistic effects, such as defects in the grid in the form of holes, and quantum properties such as interference effects. The results are eye-opening.

The question they ask is how quickly an electron can find the hole in a grid. In other words, this is evidence that free electrons naturally implement the Grover search algorithm when moving across the surface of certain crystals. That has immediate implications for quantum computing. The work also has implications for our thinking about the genetic code and the origin of life.

Every living creature on Earth uses the same code, in which DNA stores information using four nucleotide bases. The sequences of nucleotides encode information for constructing proteins from an alphabet of 20 amino acids. But why these numbers—four and 20—and not some others? In this situation, the molecular machinery inside a cell must search through the molecular soup of nucleotide bases to find the right one.

If there are four choices, a classical search takes four steps on average. So the machinery would have to try four different bases during each assembly step. Indeed, four is optimal number. This thinking also explains why there are 20 amino acids. In DNA, each set of three nucleotides defines a single amino acid. So the sequence of triplets in DNA defines the sequence of amino acids in a protein.

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But during protein assembly, each amino acid must be chosen from a soup of 20 different options. Again, 20 is the optimal number. Thus, while this equivalency does not provide any help in solving the problem on a classical computer since it would need to generate this sequence of 2 n numbers for an n-bit number to factor, which would take an exponential amount of time , it is a perfect problem for a quantum computer.

In a quantum computer, the exponentially long sequence can be encoded into merely n qubits, and generated in a time that is polynomial in n. Once that sequence is generated, the QFT can be used to find the period. The fact that the returned result is only a sample of the output FT amplitudes is not limiting, since the desired information is highly likely to be what a measurement would sample. In addition, it would be able to compute the.

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The implications of quantum computing for cryptography are discussed in more detail in Chapter 4. Currently, there are quantum approaches for solving some cases of this problem efficiently, specifically for so-called Abelian and closely related groups characterized by their symmetry properties.

On the other hand, the problem is expected to be hard for the so-called dihedral symmetry group. This hard problem is closely related to another, called the shortest vector problem—the basis of the learning with errors LWE cryptosystem, one of the proposed post-quantum that, is, quantum-resistant cyphers described in Chapter 4.

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While this is only a polynomial speedup over the best classical approach, it could nonetheless be significant in practice. As will be discussed in the next chapter, this could be sufficient to compromise additional cryptographic operations. Moreover, this is the optimal quantum algorithm for this problem, in light of the result of Bennett et al.

The problem with the classical exhaustive search approach is that the systematic testing of each possible answer is a blind guess-and-check: each query provides no information about the answer until it is actually found. The first effectively flags the state corresponding to the correct answer by changing the sign of its coefficient. The second, called the Grover diffusion operator, is then able to slightly increase the magnitude of this coefficient. Together, the two steps comprise a so-called Grover iteration, each application of which increases the probability that the correct answer will be read-out upon measurement.

This procedure to increase the amplitude of the state s containing the correct answer is an example of a general algorithmic approach called amplitude amplification [ 16 ], which is useful in a number of quantum algorithms.

A classical random walk approach can explore an area proportional to the square root of the number of steps; the quantum random walk can explore an area proportional to the number of steps. Hence, the quantum algorithm provides quadratic speedup. This technique is very versatile and has led to a number of quantum algorithms providing polynomial speedups for specific computational tasks. For example, there is a quantum walk-based algorithm for solving the basic problem of determining whether the player making the first move has a winning strategy in a combinatorial game such as chess.

More generally, the quantum algorithm provides a quadratic speedup for evaluating any AND-OR formula [ 17 , 18 ]. To perform a true quantum search, the set of data to be searched must first be represented as a superposition of quantum states and for a quantum algorithm to provide any speedup, this representation would need to be created in a time much less than the number of data points, N —somewhere between O 1 to O log N. In the classical case, this data would simply be stored in random access memory RAM and called when needed.

However, while RAM is a key element of classical computing, there is currently no robust practical RAM equivalent that generates the needed quantum superposition state for a quantum computer. It has been proposed that a quantum version of random access memory RAM , or QRAM, could generate this data in O log N time [ 19 ], although this has not been practically demonstrated. To achieve this, a classical data storage unit would be supplemented with quantum logic.

A classical analogue to this structure, called a content addressable memory, or CAM, exists, and solves this search problem in O log N time. However, with CAM and with QRAM, getting the data into the device in the first place still takes O N time, so either approach will only be useful when multiple queries are performed on the same data set—that is, the utility of CAM and QRAM, if it can be built, grows in direct proportion to the number of times the input can be reused.

Quantum algorithms can exponentially outperform classical ones when simulating a system with many quantum degrees of freedom, with applications including problems in chemistry, materials science, condensed matter, nuclear physics, and high-energy physics. The general objective in simulating a quantum system is to determine its structure or behavior, given knowledge of its components and the environment in which it exists.

For example, a simulation could be used to elucidate the structure of a substance, or the behavior over time of a collection of interacting particles. These problems can have a variety of applications, from informing the development of new industrial materials to solving important physics problems. In general, these simulations require knowledge of the Hamiltonian energy operator describing all elements and interactions of the system. From there, one can either solve for the ground-state wave function for that system in the time-independent picture , or, given some initial state of the system at a time t 0 , compute a close approximation to the quantum state at a future time t.

Scientists have been performing classical simulations of quantum systems for decades, either restricting attention to small systems or relying upon approximate methods that can trade accuracy for computational efficiency. Accurate models are so compute-intensive given the intrinsic high dimensionality of quantum systems as to be inadequate for all but small systems. A quantum, rather than classical, simulation is naturally better equipped to explore the state space spanned by quantum systems.

In principle, quantum simulation can proceed via at least three general approaches, each of which promises more efficient solutions in certain scenarios. Last, in the field of analog quantum simulation, dedicated quantum systems, although not full-blown. While this hardware is likely to be much simpler than gate-based machine solving the same problem, the downside of the analog simulation approach is that the hardware has limitations on the Hamiltonians it can create, so the resulting system is special-purpose and the application and simulator need to be co-designed.

In addition, unlike digital quantum computations that can be protected using fault-tolerant protocols, the ability to perform analog quantum simulation in realistic, noisy environments is less well understood. In the time-evolution Hamiltonian simulation algorithms, the form of the Hamiltonian, and potentially its own dependence on time, as well as the initial state of the system must be provided as inputs.

The algorithm begins by initializing the qubits into the initial system state or an approximation to it. In practice, the overall Hamiltonian is usually represented as a sum of smaller, so-called local Hamiltonians, each of which act on only a component of the larger system, which provides a useful decomposition more generally, the Hamiltonian can be simulated efficiently provided it is sparse and the nonzero entries in any given row can be efficiently located and computed.

### Introduction

For the process to proceed efficiently, the method of encoding of the initial state in qubits and of representing the time propagation as a gate sequence must be carefully chosen for the system in question. The first concrete quantum algorithms for gate-based Hamiltonian simulation were developed in the mids [ 21 ], and additional methods for different kinds of quantum systems has followed, along with algorithmic insights that have yielded significant reductions in time [ 22 - 28 ]. Efficient Hamiltonian simulation on a quantum computer would enable important speedups for problems in quantum chemistry and materials simulation [ 29 , 30 ].

In particular, the electron correlation problem has been one of the most challenging problems for classical methods to tackle [ 31 ]. To understand and predict complex reaction mechanisms involved in, for example, a transition-metal catalyzed chemical transformation, requires extremely accurate electronic structure approaches.

Classically, even molecules with fewer than one hundred strongly correlated electrons are beyond the scale of classical ab initio methods at chemical accuracy. Quantum computers promise exponential speedups for the simulation of the electronic structure problem and it has been shown that they would enable efficient elucidation of chemical reaction mechanisms [ 32 ]. Childs, , On the relationship between continuous- and discrete-time quantum walk, Communications in Mathematical Physics 2 Strongly correlated species involved in chemical reactions where classical approaches presently fail include problems such as photochemical processes, nitrogen fixation, C-H bond breaking, carbon dioxide fixation and transformation, hydrogen and oxygen production, and other transition-metal catalysis problems.

These applications extend to important industrial applications including fertilizer production, polymerization catalysis and clean energy processes. Hamiltonian simulation is also used within quantum algorithms for solving complex correlated material problems [ 35 ], which may have application in, for example, the search for a high-temperature superconductor. Quantum computers promise an exponential speedup over classical approaches for time evolution of quantum systems. Thus, quantum computers may have the most impact in their application to problems in quantum chemistry, for example as applied in pharmaceutics, and materials science [ 36 ].

However, there are many types of Hamiltonians that will require new methods if they are to become efficiently solvable on a quantum computer. For example, to model the electronic structure for applications in quantum chemistry [ 37 ], the Hamiltonian of an n -orbital system involves O n 4 terms, which means a low-error quantum computer will be required for its computation.

Researchers have recently combined these techniques with the existing framework for quantum Hamiltonian simulation that has led to rapid progress in these quantum algorithms for such problems [ 39 - 47 ]. Hamiltonian simulation has also proven to be a powerful tool for developing quantum algorithms for problems with no immediate connection to quantum mechanics. A prominent example is the recent development of a new class of quantum algorithms that directly perform linear algebra on the quantum state; this is discussed next.

Linear algebra, a foundational area of mathematics, can be useful in a range of contexts, from the science of quantum mechanics to computer graphics design to methods in machine learning. The general task of linear algebra is to find the solution to a system of linear equations, that is, one or more equations of the form where the sum of a set of independent variables, each scaled by some coefficient, is equal to a constant value.

A quantum algorithm for such applications, termed HHL after its developers Harrow, Hassidim, and Lloyd, makes use of methods from Hamiltonian simulation [ 48 ]. It also assumes that the matrix A is sparse and its entries are accessible via an easy to compute function. This is a procedure for exponentially fast estimation of the eigenvalue or phase of an eigenvector of a unitary operator.

This is relevant to linear algebra, since inverting the matrix A is easy if its eigenvalues are known. For a given A and b, the algorithm would output a quantum state for which the values of the N coefficients are proportional to the N elements of solution x. Although the solution is present in the quantum computer, the rules of quantum mechanics also prevent it from being directly read out. However, if one is interested in only finding certain expectation values of the solution, one can obtain this result with a number of gates that has poly log N cost [ 49 ].

Linear algebra problems can be solved with a classical computer using memory and running time that scale as poly N so a quantum computer would use exponentially fewer resources and time for solving this more restricted problem. As with the preceding algorithms, this exponential speedup comes with a number of important caveats. As previously mentioned, reading the output provides only an index i with probability proportional to x i 2.

Thus, one major issue in using this algorithm is finding settings where this limited information is useful.