This paper explores advancements in quantum algorithms for derivative pricing of exotics, a computational pipeline of fundamental importance in quantitative finance. For such cases, the classical Monte Carlo integration procedure provides the state-of-the-art provable, asymptotic performance: polynomial in problem dimension and quadratic in inverse-precision. While quantum algorithms are known to offer quadratic speedups over classical Monte Carlo methods, end-to-end speedups have been proven only in the simplified setting over the Black-Scholes geometric Brownian motion (GBM) model. This paper extends existing frameworks to demonstrate novel quadratic speedups for more practical models, such as the Cox-Ingersoll-Ross (CIR) model and a variant of Heston's stochastic volatility model, utilizing a characteristic of the underlying SDEs which we term fast-forwardability. Additionally, for general models that do not possess the fast-forwardable property, we introduce a quantum Milstein sampler, based on a novel quantum algorithm for sampling Lévy areas, which enables quantum multi-level Monte Carlo to achieve quadratic speedups for multi-dimensional stochastic processes exhibiting certain correlation types. We also present an improved analysis of numerical integration for derivative pricing, leading to substantial reductions in the resource requirements for pricing GBM and CIR models. Furthermore, we investigate the potential for additional reductions using arithmetic-free quantum procedures. Finally, we critique quantum partial differential equation (PDE) solvers as a method for derivative pricing based on amplitude estimation, identifying theoretical barriers that obstruct achieving a quantum speedup through this approach. Our findings significantly advance the understanding of quantum algorithms in derivative pricing, addressing key challenges and open questions in the field.

On Tuesday 26th November 2024, four discussants participated in a moderated virtual panel titled Future of Quantum Computing as one session of the 8th International Conference on Quantum Techniques in Machine Learning hosted by the University of Melbourne. This article provides a detailed summary of the discussion in this lively session.

Free energy calculations are at the heart of physics-based analyses of biochemical processes. They allow us to quantify molecular recognition mechanisms, which determine a wide range of biological phenomena, from how cells send and receive signals to how pharmaceutical compounds can be used to treat diseases. Quantitative and predictive free energy calculations require computational models that accurately capture both the varied and intricate electronic interactions between molecules as well as the entropic contributions from the motions of these molecules and their aqueous environment. However, accurate quantum-mechanical energies and forces can be obtained only for small atomistic models and not for large biomacromolecules. Here, we demonstrate how to consistently link accurate quantum-mechanical data obtained for substructures to the overall potential energy of biomolecular complexes using machine learning in an integrated algorithm. We do so using a two-fold quantum embedding strategy where the innermost quantum cores are treated at a very high level of accuracy. We demonstrate the viability of this approach for the molecular recognition of a ruthenium-based anticancer drug by its protein target by applying traditional quantum chemical methods. As such methods scale unfavorably with system size, we analyze the requirements for quantum computers to provide highly accurate energies that affect the resulting free energies. Once the requirements are met, our computational pipeline, FreeQuantum, is able to make efficient use of the quantum-computed energies, thereby enabling quantum computing-enhanced modeling of biochemical processes. This approach combines the exponential speedups of quantum computers for simulating interacting electrons with modern classical simulation techniques that incorporate machine learning to model large molecules.

A fundamental task in quantum information is to approximate a pure quantum state in terms of sparse states or, for a bipartite system, states of bounded Schmidt rank. The optimal deterministic approximation in each case is straightforward, and maximizes the fidelity: keep the largest entries or singular values. On the other hand, random mixtures of sparse states can achieve quadratically improved trace distances, and yield nontrivial bounds on other distance measures like the robustness. In this work, we give efficient algorithms for finding mixtures of sparse states that optimally approximate a given pure state in either trace distance or robustness. These algorithms also yield descriptions of efficiently samplable ensembles of sparse, or less-entangled, states that correspond to these optimal mixed approximations. This can be used for the truncation step of algorithms for matrix product states, improving their accuracy while using no extra memory, and we demonstrate this improvement numerically. Our proofs use basic facts about convex optimization and zero-sum games, as well as rigorous guarantees for computing maximum-entropy distributions.

In an isolated quantum many-body system undergoing unitary evolution, the entropy of a subsystem (smaller than half the system size) thermalizes if at long times, it is to leading order equal to the thermodynamic entropy of the subsystem at the same energy. In this paper, we prove entropy thermalization for a nearly integrable Sachdev-Ye-Kitaev model initialized in a pure product state. The model is obtained by adding random all-to-all 4-body interactions as a perturbation to a random free-fermion model.

We present a case study and forward-looking perspective on co-design for hybrid quantum-classical algorithms, centered on the goal of empirical quantum advantage (EQA), which we define as a measurable performance gain using quantum hardware over state-of-the-art classical methods on the same task. Because classical algorithms continue to improve, the EQA crossover point is a moving target; nevertheless, we argue that a persistent advantage is possible for our application class even if the crossover point shifts. Specifically, our team examines the task of biomarker discovery in precision oncology. We push the limitations of the best classical algorithms, improving them as best as we can, and then augment them with a quantum subroutine for the task where we are most likely to see performance gains. We discuss the implementation of a quantum subroutine for feature selection on current devices, where hardware constraints necessitate further co-design between algorithm and physical device capabilities. Looking ahead, we perform resource analysis to explore a plausible EQA region on near/intermediate-term hardware, considering the impacts of advances in classical and quantum computing on this regime. Finally, we outline potential clinical impact and broader applications of this hybrid pipeline beyond oncology.

We study methods to replace entangling operations with random local operations in a quantum computation, at the cost of increasing the number of required executions. First, we consider “spacelike cuts” where an entangling unitary is replaced with random local unitaries. We propose an entanglement measure for quantum dynamics, the product extent, which bounds the cost in a procedure for this replacement based on two copies of the Hadamard test. In the terminology of prior work, this procedure yields a quasiprobability decomposition with minimal 1-norm in a number of cases, which addresses an open question of Piveteau and Sutter. As an application, we give an improved algorithm for clustered Hamiltonian simulation. Specifically we show that interactions can be removed at a cost, which is exponential in the sum of their strengths times the evolution time, and vanishing in the limit of weak interactions. We also give an improved upper bound on the cost of replacing wires with measure-and-prepare channels using “timelike cuts.” We prove a matching information-theoretic lower bound when estimating output probabilities.

There are few known exponential speedups for quantum algorithms and these tend to fall into even fewer families. One speedup that has mostly resisted generalization is the use of quantum walks to traverse the welded-tree graph, due to Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman. We show how to generalize this to a large class of hierarchical graphs in which the vertices are grouped into "supervertices" which are arranged according to a d-dimensional lattice. Supervertices can have different sizes, and edges between supervertices correspond to random connections between their constituent vertices. The hitting times of quantum walks on these graphs are related to the localization properties of zero modes in certain disordered tight binding Hamiltonians. The speedups range from superpolynomial to exponential, depending on the underlying dimension and the random graph model. We also provide concrete realizations of these hierarchical graphs, and introduce a general method for constructing graphs with efficient quantum traversal times using graph sparsification.

We consider the asymptotic limits where certain parameters in the definitions of the Laguerre and Jacobi ensembles diverge. In these limits, Dette, Imhof, and Nagel proved that up to a linear transformation, the joint probability distributions of the ensembles become more and more concentrated around the zeros of the Laguerre and Jacobi polynomials, respectively. In this paper, we improve the concentration bounds. Our proofs are similar to those in the original references, but the error analysis is improved and arguably simpler. For the first and second moments of the Jacobi ensemble, we further improve the concentration bounds implied by our aforementioned results.

Consider the n! different unitary matrices that permute n d-dimensional quantum systems. If d≥ n then they are linearly independent. This paper discusses a sense in which they are approximately orthogonal (with respect to the Hilbert–Schmidt inner product, ⟨ A,B⟩ = trA^† B/trI ) if d≫ n^2 , or, in a different sense, if d≫ n . Previous work had shown pairwise approximate orthogonality of these matrices, but here we show a more collective statement, quantified in terms of the operator norm distance of the Gram matrix to the identity matrix. This simple point has several applications in quantum information and random matrix theory: (1) showing that random maximally entangled states resemble fully random states, (2) showing that Boson sampling output probabilities resemble those from Gaussian matrices, (3) improving the Eggeling–Werner scheme for multipartite data hiding, (4) proving that the product test of Harrow–Montanaro cannot be performed using LOCC without a large number of copies of the state to be tested, (5) proving that the purity of a quantum state also cannot be efficiently tested using LOCC, and (6, published separately with Brandão and Horodecki) helping prove that poly-size random quantum circuits are poly-designs.

We prove that $poly(t) \cdot n^{1/D}$-depth local random quantum circuits with two qudit nearest-neighbor gates on a $D$-dimensional lattice with n qudits are approximate $t$-designs in various measures. These include the "monomial" measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was $poly(t)\cdot n$ due to Brandao-Harrow-Horodecki (BHH) for $D=1$. We also improve the "scrambling" and "decoupling" bounds for spatially local random circuits due to Brown and Fawzi. One consequence of our result is that assuming the polynomial hierarchy (PH) is infinite and that certain counting problems are $\#P$-hard on average, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under the assumption that PH is infinite. However, to show the hardness of approximate sampling using this strategy requires that the quantum circuits have a property called "anti-concentration", meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Thus our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent proposal by the Google Quantum AI group to perform such a sampling task with 49 qubits on a two-dimensional lattice and confirms their conjecture that $O(\sqrt n)$ depth suffices for anti-concentration. We also prove that anti-concentration is possible in depth O(log(n) loglog(n)) using a different model.

Quantum information theory seeks to understand the fundamental limits set by quantum mechanics for information processing tasks. The mathematical aspects of quantum information rely on tools from various fields including mathematical optimization, high-dimensional convex geometry, operator algebras and representation theory. The goal of this meeting is to focus on the mathematical aspects connecting geometry, optimization and quantum information theory and develop new tools to solve some of the open problems at the intersection of these fields.

Ground-state entanglement governs various properties of quantum many-body systems at low temperatures and is the key to understanding gapped quantum phases of matter. Here we identify a structural property of entanglement in the ground state of gapped local Hamiltonians. This property is captured using a quantum information quantity known as the entanglement spread, which measures the difference between Rényi entanglement entropies. Our main result shows that gapped ground states possess limited entanglement spread across any partition of the system, exhibiting an area-law scaling. Our result applies to systems with interactions described by any graph, but we obtain an improved bound for the special case of lattices. These interaction graphs include cases where entanglement entropy is known not to satisfy an area law. We achieve our results first by connecting the ground-state entanglement to the communication complexity of testing bipartite entangled states and then devising a communication scheme for testing ground states using recently developed quantum algorithms for Hamiltonian simulation. Entanglement entropy between two parts of a quantum state generally grows with volume, but for one-dimensional and some two-dimensional ground states, it scales with area. An area law has now been proven for a related metric in any dimension or geometry.

In an isolated quantum many-body system undergoing unitary evolution, we study the thermalization of a subsystem, treating the rest of the system as a bath. In this setting, the eigenstate thermalization hypothesis (ETH) was proposed to explain thermalization. Consider a nearly integrable Sachdev-Ye-Kitaev model obtained by adding random all-to-all 4-body interactions as a perturbation to a random free-fermion model. When the subsystem size is larger than the square root of but is still a vanishing fraction of the system size, we prove thermalization if the system is initialized in a random product state, while almost all eigenstates violate the ETH. In this sense, the ETH is not a necessary condition for thermalization.

A global quantum quench can be modeled by a quantum circuit with local unitary gates. In general, entanglement grows linearly at a rate given by entanglement velocity, which is upper bounded by the growth of the light cone. We show that the unitary interactions achieving the maximal rate must remain unitary if we exchange the space and time directions – a property known as dual unitarity. Our results are robust: approximate maximal entanglement velocity also implies approximate dual unitarity. We further show that maximal entanglement velocity is always accompanied by a specific dynamical pattern of entanglement, which yields simpler analyses of several known exactly solvable models.

Random quantum circuits are commonly viewed as hard to simulate classically. In some regimes this has been formally conjectured, and there had been no evidence against the more general possibility that for circuits with uniformly random gates, approximate simulation of typical instances is almost as hard as exact simulation. We prove that this is not the case by exhibiting a shallow circuit family with uniformly random gates that cannot be efficiently classically simulated near-exactly under standard hardness assumptions, but can be simulated approximately for all but a superpolynomially small fraction of circuit instances in time linear in the number of qubits and gates. We furthermore conjecture that sufficiently shallow random circuits are efficiently simulable more generally. To this end, we propose and analyze two simulation algorithms. Implementing one of our algorithms numerically, we give strong evidence that it is efficient both asymptotically and, in some cases, in practice. To argue analytically for efficiency, we reduce the simulation of 2D shallow random circuits to the simulation of a form of 1D dynamics consisting of alternating rounds of random local unitaries and weak measurements -- a type of process that has generally been observed to undergo a phase transition from an efficient-to-simulate regime to an inefficient-to-simulate regime as measurement strength is varied. Using a mapping from quantum circuits to statistical mechanical models, we give evidence that a similar computational phase transition occurs for our algorithms as parameters of the circuit architecture like the local Hilbert space dimension and circuit depth are varied.

The out-of-time-ordered correlation (OTOC) and entanglement are two physically motivated and widely used probes of the"scrambling"of quantum information, a phenomenon that has drawn great interest recently in quantum gravity and many-body physics. We argue that the corresponding notions of scrambling can be fundamentally different, by proving an asymptotic separation between the time scales of the saturation of OTOC and that of entanglement entropy in a random quantum circuit model defined on graphs with a tight bottleneck, such as tree graphs. Our result counters the intuition that a random quantum circuit mixes in time proportional to the diameter of the underlying graph of interactions. It also provides a more rigorous justification for an argument in our previous work arXiv:1807.04363, that black holes may be slow information scramblers, which in turn relates to the black hole information problem. The bounds we obtained for OTOC are interesting in their own right in that they generalize previous studies of OTOC on lattices to the geometries on graphs in a rigorous and general fashion.

We propose an efficient heuristic for mapping the logical qubits of quantum algorithms to the physical qubits of connectivity-limited devices, adding a minimal number of connectivity-compliant SWAP gates. In particular, given a quantum circuit, we construct an undirected graph with edge weights a function of the two-qubit gates of the quantum circuit. Taking inspiration from spectral graph drawing, we use an eigenvector of the graph Laplacian to place logical qubits at coordinate locations. These placements are then mapped to physical qubits for a given connectivity. We primarily focus on one-dimensional connectivities and sketch how the general principles of our heuristic can be extended for use in more general connectivities.

Computing plays a critical role in the biological sciences but faces increasing challenges of scale and complexity. Quantum computing, a computational paradigm exploiting the unique properties of quantum mechanical analogs of classical bits, seeks to address many of these challenges. We discuss the potential for quantum computing to aid in the merging of insights across different areas of biological sciences.

The authors prove an inequality usable to certify the presence of entanglement in noisy macroscopic systems. Violations of this inequality could potentially be observed experimentally.