Quantum 7, 1191 (2023).

Quantum machine learning has the potential for a transformative impact across industry sectors and in particular in finance. In our work we look at the problem of hedging where deep reinforcement learning offers a powerful framework for real markets. We develop quantum reinforcement learning methods based on policy-search and distributional actor-critic algorithms that use quantum neural network architectures with orthogonal and compound layers for the policy and value functions. We prove that the quantum neural networks we use are trainable, and we perform extensive simulations that show that quantum models can reduce the number of trainable parameters while achieving comparable performance and that the distributional approach obtains better performance than other standard approaches, both classical and quantum. We successfully implement the proposed models on a trapped-ion quantum processor, utilizing circuits with up to $16$ qubits, and observe performance that agrees well with noiseless simulation. Our quantum techniques are general and can be applied to other reinforcement learning problems beyond hedging.

Quantum 7, 1190 (2023).

We develop a general perturbative theory of finite-coupling quantum thermometry up to second order in probe-sample interaction. By assumption, the probe and sample are in thermal equilibrium, so the probe is described by the mean-force Gibbs state. We prove that the ultimate thermometric precision can be achieved – to second order in the coupling – solely by means of local energy measurements on the probe. Hence, seeking to extract temperature information from coherences or devising adaptive schemes confers no practical advantage in this regime. Additionally, we provide a closed-form expression for the quantum Fisher information, which captures the probe's sensitivity to temperature variations. Finally, we benchmark and illustrate the ease of use of our formulas with two simple examples. Our formalism makes no assumptions about separation of dynamical timescales or the nature of either the probe or the sample. Therefore, by providing analytical insight into both the thermal sensitivity and the optimal measurement for achieving it, our results pave the way for quantum thermometry in setups where finite-coupling effects cannot be ignored.

Quantum 7, 1189 (2023).

In light of recently proposed quantum algorithms that incorporate symmetries in the hope of quantum advantage, we show that with symmetries that are restrictive enough, classical algorithms can efficiently emulate their quantum counterparts given certain classical descriptions of the input. Specifically, we give classical algorithms that calculate ground states and time-evolved expectation values for permutation-invariant Hamiltonians specified in the symmetrized Pauli basis with runtimes polynomial in the system size. We use tensor-network methods to transform symmetry-equivariant operators to the block-diagonal Schur basis that is of polynomial size, and then perform exact matrix multiplication or diagonalization in this basis. These methods are adaptable to a wide range of input and output states including those prescribed in the Schur basis, as matrix product states, or as arbitrary quantum states when given the power to apply low depth circuits and single qubit measurements.

Quantum 7, 1188 (2023).

Previously proposed quantum algorithms for solving linear systems of equations cannot be implemented in the near term due to the required circuit depth. Here, we propose a hybrid quantum-classical algorithm, called Variational Quantum Linear Solver (VQLS), for solving linear systems on near-term quantum computers. VQLS seeks to variationally prepare $|x\rangle$ such that $A|x\rangle\propto|b\rangle$. We derive an operationally meaningful termination condition for VQLS that allows one to guarantee that a desired solution precision $\epsilon$ is achieved. Specifically, we prove that $C \geqslant \epsilon^2 / \kappa^2$, where $C$ is the VQLS cost function and $\kappa$ is the condition number of $A$. We present efficient quantum circuits to estimate $C$, while providing evidence for the classical hardness of its estimation. Using Rigetti's quantum computer, we successfully implement VQLS up to a problem size of $1024\times1024$. Finally, we numerically solve non-trivial problems of size up to $2^{50}\times2^{50}$. For the specific examples that we consider, we heuristically find that the time complexity of VQLS scales efficiently in $\epsilon$, $\kappa$, and the system size $N$.

Quantum 7, 1187 (2023).

Single photon detector(SPD) problems arise in most quantum tasks, especially for measuring states going through high-lost channels. They are particularly prominent in quantum key distribution(QKD), which could be the most significant application in quantum information theory. In recent years, QKD distance has been improved dramatically but is still restricted because the bit error rate(QBER) caused by SPD dark counts will be out of control as the distance increases. If this problem can be solved, QKD can be implemented over arbitrarily long distances. However, previous solutions often result in impractical requirements such as superconductors while they can only reduce the dark count rate to finite low levels. In this paper, we solve SPD problems with today's technologies only. Although it is the no-cloning theorem that prevents a state from being measured multiple times to obtain a more reliable result, we propose a scheme circumventing the no-cloning theorem in certain tasks to allow a single state to be employed several times. The scheme demonstrates that imperfect detectors can provide nearly perfect results, namely, the QBER caused by dark counts can be reduced to arbitrarily low while in the meantime, detective efficiency can be improved to arbitrarily high. Consequently, QKD distance is not limited by the imperfect SPD anymore and can be improved from hundreds of kilometers to thousands without high-technology detectors. Furthermore, similar schemes can be applied for reducing measurement errors or improving the performance of sources. Finally, it is worth noting that although the paper is mainly discussed in the context of QKD, our scheme is an independent scheme that could be employed in other protocols wherever SPD are employed.

Quantum 7, 1186 (2023).

Quantum algorithms are getting extremely popular due to their potential to significantly outperform classical algorithms. Yet, applying quantum algorithms to optimization problems meets challenges related to the efficiency of quantum algorithms training, the shape of their cost landscape, the accuracy of their output, and their ability to scale to large-size problems. Here, we present an approximate gradient-based quantum algorithm for hardware-efficient circuits with amplitude encoding. We show how simple linear constraints can be directly incorporated into the circuit without additional modification of the objective function with penalty terms. We employ numerical simulations to test it on $\texttt{MaxCut}$ problems with complete weighted graphs with thousands of nodes and run the algorithm on a superconducting quantum processor. We find that for unconstrained $\texttt{MaxCut}$ problems with more than 1000 nodes, the hybrid approach combining our algorithm with a classical solver called CPLEX can find a better solution than CPLEX alone. This demonstrates that hybrid optimization is one of the leading use cases for modern quantum devices.

Quantum 7, 1185 (2023).

Stabilizer simulation can efficiently simulate an important class of quantum circuits consisting exclusively of Clifford gates. However, all existing extensions of this simulation to arbitrary quantum circuits including non-Clifford gates suffer from an exponential runtime. To address this challenge, we present a novel approach for efficient stabilizer simulation on arbitrary quantum circuits, at the cost of lost precision. Our key idea is to compress an exponential sum representation of the quantum state into a single $abstract$ summand covering (at least) all occurring summands. This allows us to introduce an $\textit{abstract stabilizer simulator}$ that efficiently manipulates abstract summands by $over-approximating$ the effect of circuit operations including Clifford gates, non-Clifford gates, and (internal) measurements. We implemented our abstract simulator in a tool called Abstraqt and experimentally demonstrate that Abstraqt can establish circuit properties intractable for existing techniques.

Quantum 7, 1184 (2023).

We propose a quantum error mitigation strategy for the variational quantum eigensolver (VQE) algorithm. We find, via numerical simulation, that very small amounts of coherent noise in VQE can cause substantially large errors that are difficult to suppress by conventional mitigation methods, and yet our proposed mitigation strategy is able to significantly reduce these errors. The proposed strategy is a combination of previously reported techniques, namely randomized compiling (RC) and zero-noise extrapolation (ZNE). Intuitively, randomized compiling turns coherent errors in the circuit into stochastic Pauli errors, which facilitates extrapolation to the zero-noise limit when evaluating the cost function. Our numerical simulation of VQE for small molecules shows that the proposed strategy can mitigate energy errors induced by various types of coherent noise by up to two orders of magnitude.

Quantum 7, 1183 (2023).

Fault-tolerant quantum computing requires classical hardware to perform the decoding necessary for error correction. The Union–Find decoder is one of the best candidates for this. It has remarkably organic characteristics, involving the growth and merger of data structures through nearest-neighbour steps; this naturally suggests the possibility of its realisation using a lattice of simple processors with nearest-neighbour links. In this way the computational load can be distributed with near-ideal parallelism. Here we show for the first time that this strict (rather than partial) locality is practical, with a worst-case runtime $\mathcal O(d^3)$ and mean runtime subquadratic in the surface code distance $d$. A novel parity-calculation scheme is employed which can simplify previously proposed architectures, and our approach is optimised for circuit-level noise. We compare our local realisation with one augmented by long-range links; while the latter is of course faster, we note that local asynchronous logic could negate the difference.

Quantum 7, 1182 (2023).

Measurements take a singular role in quantum theory. While they are often idealized as an instantaneous process, this is in conflict with all other physical processes in nature. In this Letter, we adopt a standpoint where the interaction with an environment is a crucial ingredient for the occurrence of a measurement. Within this framework, we derive lower bounds on the time needed for a measurement to occur. Our bound scales proportionally to the change in entropy of the measured system, and decreases as the number of of possible measurement outcomes or the interaction strength driving the measurement increases. We evaluate our bound in two examples where the environment is modelled by bosonic modes and the measurement apparatus is modelled by spins or bosons.

Quantum 7, 1181 (2023).

In this paper we provide a framework for combining multiple quantum simulation methods, such as Trotter-Suzuki formulas and QDrift into a single Composite channel that builds upon older coalescing ideas for reducing gate counts. The central idea behind our approach is to use a partitioning scheme that allocates a Hamiltonian term to the Trotter or QDrift part of a channel within the simulation. This allows us to simulate small but numerous terms using QDrift while simulating the larger terms using a high-order Trotter-Suzuki formula. We prove rigorous bounds on the diamond distance between the Composite channel and the ideal simulation channel and show under what conditions the cost of implementing the Composite channel is asymptotically upper bounded by the methods that comprise it for both probabilistic partitioning of terms and deterministic partitioning. Finally, we discuss strategies for determining partitioning schemes as well as methods for incorporating different simulation methods within the same framework.

Quantum 7, 1180 (2023).

We give an approximation algorithm for Quantum Max-Cut which works by rounding an SDP relaxation to an entangled quantum state. The SDP is used to choose the parameters of a variational quantum circuit. The entangled state is then represented as the quantum circuit applied to a product state. It achieves an approximation ratio of 0.582 on triangle-free graphs. The previous best algorithms of Anshu, Gosset, Morenz, and Parekh, Thompson achieved approximation ratios of 0.531 and 0.533 respectively. In addition, we study the EPR Hamiltonian, which we argue is a natural intermediate problem which isolates some key quantum features of local Hamiltonian problems. For the EPR Hamiltonian, we give an approximation algorithm with approximation ratio $1 / \sqrt{2}$ on all graphs.

Quantum 7, 1179 (2023).

This work provides a relativistic, digital quantum simulation scheme for both $2+1$ and $3+1$ dimensional quantum electrodynamics (QED), based on a discrete spacetime formulation of theory. It takes the form of a quantum circuit, infinitely repeating across space and time, parametrised by the discretization step $\Delta_t=\Delta_x$. Strict causality at each step is ensured as circuit wires coincide with the lightlike worldlines of QED; simulation time under decoherence is optimized. The construction replays the logic that leads to the QED Lagrangian. Namely, it starts from the Dirac quantum walk, well-known to converge towards free relativistic fermions. It then extends the quantum walk into a multi-particle sector quantum cellular automata in a way which respects the fermionic anti-commutation relations and the discrete gauge invariance symmetry. Both requirements can only be achieved at cost of introducing the gauge field. Lastly the gauge field is given its own electromagnetic dynamics, which can be formulated as a quantum walk at each plaquette.

Quantum 7, 1178 (2023).

Security of a storage device against a tampering adversary has been a well-studied topic in classical cryptography. Such models give black-box access to an adversary, and the aim is to protect the stored message or abort the protocol if there is any tampering. In this work, we extend the scope of the theory of tamper detection codes against an adversary with quantum capabilities. We consider encoding and decoding schemes that are used to encode a $k$-qubit quantum message $\vert m\rangle$ to obtain an $n$-qubit quantum codeword $\vert {\psi_m} \rangle$. A quantum codeword $\vert {\psi_m} \rangle$ can be adversarially tampered via a unitary $U$ from some known tampering unitary family $\mathcal{U}_{\mathsf{Adv}}$ (acting on $\mathbb{C}^{2^n}$). Firstly, we initiate the general study of $\textit{quantum tamper detection codes}$, which detect if there is any tampering caused by the action of a unitary operator. In case there was no tampering, we would like to output the original message. We show that quantum tamper detection codes exist for any family of unitary operators $\mathcal{U}_{\mathsf{Adv}}$, such that $\vert\mathcal{U}_{\mathsf{Adv}} \vert \lt 2^{2^{\alpha n}}$ for some constant $\alpha \in (0,1/6)$; provided that unitary operators are not too close to the identity operator. Quantum tamper detection codes that we construct can be considered to be quantum variants of $\textit{classical tamper detection codes}$ studied by Jafargholi and Wichs ['15], which are also known to exist under similar restrictions. Additionally, we show that when the message set $\mathcal{M}$ is classical, such a construction can be realized as a $\textit{non-malleable code}$ against any $\mathcal{U}_{\mathsf{Adv}}$ of size up to $2^{2^{\alpha n}}$.

Quantum 7, 1177 (2023).

The role of entanglement in determining the non-classicality of a given interaction has gained significant traction over the last few years. In particular, as the basis for new experimental proposals to test the quantum nature of the gravitational field. Here we show that the rate of gravity mediated entanglement between two otherwise isolated optomechanical systems can be significantly increased by modulating the optomechanical coupling. This is most pronounced for low mass, high frequency systems – convenient for reaching the quantum regime – and can lead to improvements of several orders of magnitude, as well as a broadening of the measurement window. Nevertheless, significant obstacles still remain. In particular, we find that modulations increase decoherence effects at the same rate as the entanglement improvements. This adds to the growing evidence that the constraint on noise (acting on the position d.o.f) depends only on the particle mass, separation, and temperature of the environment and cannot be improved by novel quantum control. Finally, we highlight the close connection between the observation of quantum correlations and the limits of measurement precision derived via the Cramér-Rao Bound. An immediate consequence is that probing superpositions of the gravitational field places similar demands on detector sensitivity as entanglement verification.

Quantum 7, 1176 (2023).

The increasing capabilities of quantum computing hardware and the challenge of realizing deep quantum circuits require fully automated and efficient tools for compiling quantum circuits. To express arbitrary circuits in a sequence of native gates specific to the quantum computer architecture, it is necessary to make algorithms portable across the landscape of quantum hardware providers. In this work, we present a compiler capable of transforming and optimizing a quantum circuit targeting a shuttling-based trapped-ion quantum processor. It consists of custom algorithms set on top of the quantum circuit framework Pytket. The performance was evaluated for a wide range of quantum circuits and the results show that the gate counts can be reduced by factors up to 5.1 compared to standard Pytket and up to 2.2 compared to standard Qiskit compilation.

Quantum 7, 1175 (2023).

A promising approach for scaling-up trapped-ion quantum computer platforms is by storing multiple trapped-ion qubit sets ('ion crystals') in segmented microchip traps and to interconnect these via physical movement of the ions ('shuttling'). Already for realizing quantum circuits with moderate complexity, the design of suitable qubit assignments and shuttling schedules require automation. Here, we describe and test algorithms which address exactly these tasks. We describe an algorithm for fully automated generation of shuttling schedules, complying to constraints imposed by a given trap structure. Furthermore, we introduce different methods for initial qubit assignment and compare these for random circuit (of up to 20 qubits) and quantum Fourier transform-like circuits, and generalized Toffoli gates of up to 40 qubits each. We find that for quantum circuits which contain a fixed structure, advanced assignment algorithms can serve to reduce the shuttling overhead.

Quantum 7, 1174 (2023).

At the interface of machine learning and quantum computing, an important question is what distributions can be learned provably with optimal sample complexities and with quantum-accelerated time complexities. In the classical case, Klivans and Goel discussed the $Alphatron$, an algorithm to learn distributions related to kernelized regression, which they also applied to the learning of two-layer neural networks. In this work, we provide quantum versions of the Alphatron in the fault-tolerant setting. In a well-defined learning model, this quantum algorithm is able to provide a polynomial speedup for a large range of parameters of the underlying concept class. We discuss two types of speedups, one for evaluating the kernel matrix and one for evaluating the gradient in the stochastic gradient descent procedure. We also discuss the quantum advantage in the context of learning of two-layer neural networks. Our work contributes to the study of quantum learning with kernels and from samples.

Quantum 7, 1173 (2023).

We consider the computational task of sampling a bit string $x$ from a distribution $\pi(x)=|\langle x|\psi\rangle|^2$, where $\psi$ is the unique ground state of a local Hamiltonian $H$. Our main result describes a direct link between the inverse spectral gap of $H$ and the mixing time of an associated continuous-time Markov Chain with steady state $\pi$. The Markov Chain can be implemented efficiently whenever ratios of ground state amplitudes $\langle y|\psi\rangle/\langle x|\psi\rangle$ are efficiently computable, the spectral gap of $H$ is at least inverse polynomial in the system size, and the starting state of the chain satisfies a mild technical condition that can be efficiently checked. This extends a previously known relationship between sign-problem free Hamiltonians and Markov chains. The tool which enables this generalization is the so-called fixed-node Hamiltonian construction, previously used in Quantum Monte Carlo simulations to address the fermionic sign problem. We implement the proposed sampling algorithm numerically and use it to sample from the ground state of Haldane-Shastry Hamiltonian with up to 56 qubits. We observe empirically that our Markov chain based on the fixed-node Hamiltonian mixes more rapidly than the standard Metropolis-Hastings Markov chain.