Gate teleportation allows us to implement a nonlocal unitary using local operations, classical communication (LOCC), and a shared entangled state. Known deterministic teleportation protocols consume at least one full ebit and achieve optimal entanglement consumption only for Clifford gates. Here, we present a deterministic LOCC protocol for implementing the two-qubit controlled-phase gate U_ϕ=diag(1,1,1,e^i ϕ) with ϕ∈ [0,π] whose entanglement consumption is close to optimal for every ϕ. In particular, vanishing rotation angles require vanishing entanglement.

Information-theoretic techniques are based on the assumption that resources are well characterized by independent and identically distributed (iid) states. This assumption cannot be justified operationally, since, for example, correlations between subsequent systems emitted by a source cannot be detected by any practical tomographic protocol. Operationally motivated symmetry assumptions still imply, via de Finetti theorems, that the resources are described by almost-iid states. This raises the question: Are almost-iid resources as effective as perfect iid resources for information-processing tasks? Here we address this question and prove that the conditional entropy of almost-iid states asymptotically coincides with that of iid states. As an application, this implies that squashed entanglement is robust for almost-iid states, asymptotically matching its value on iid states.

The generalized quantum Stein's lemma provides an explicit expression for the optimal error exponent when distinguishing many independent and identically distributed (iid) copies of a given bipartite state from the set of separable bipartite states. Here we prove that this result is robust, in the sense that the iid assumption can be relaxed to almost-iid. In particular, our result shows that the original argument of Brandão and Plenio, which contains a logical gap, can be made rigorous. Our proof relies on a novel continuity bound for the relative entropy of entanglement with respect to the quantum Wasserstein distance. Combined with a recent insight that almost-iid states and their exact iid counterparts are asymptotically close in this distance, the bound implies that their relative entropies of entanglement coincide asymptotically.

Uhlmann's theorem states that, for any two quan-tum states rho ABand sigma A, there exists an extension sigma ABof sigma Asuchthat the fidelity between rho AB and sigma AB equals the fidelity between their reduced states rho A and sigma A. In this work, we generalize Uhlmann's theorem to alpha-R enyi relative entropies for alpha is an element of[1/2,infinity],a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to alpha=1/2,alpha=1, and alpha=infinity, respectively.

The approximate quantum Fourier transform (AQFT) on n qubits can be implemented in logarithmic depth using 8n qubits with all-to-all connectivity, as shown in [Hales, PhD Thesis Berkeley, 2002]. However, realizing the required all-to-all connectivity can be challenging in practice. In this work, we use dynamic circuits, i.e., mid-circuit measurements and feed-forward operations, to implement the AQFT in logarithmic depth using only 4n qubits arranged on a line with nearest-neighbor connectivity. Furthermore, for states with a specific structure, the number of qubits can be further reduced to 2n while keeping the logarithmic depth and line connectivity. As part of our construction, we introduce a new implementation of an adder with logarithmic depth on a line, which allows us to improve the AQFT construction of Hales.

We correct an error in the proof of Theorem 11 in our paper “Approximate Degradable Quantum Channels”, IEEE Trans. Inf. Theory, vol. 63, no. 12, pp. 7832-7844, 2017, concerning an upper bound on the private capacity of an approximate anti-degradable channel. Furthermore, we show how to obtain a tighter bound for the quantum capacity.

Quasiprobabilistic cutting techniques allow us to partition large quantum circuits into smaller subcircuits by replacing non-local gates with probabilistic mixtures of local gates. The cost of this method is a sampling overhead that scales exponentially in the number of cuts. It is crucial to determine the minimal cost for gate cutting and to understand whether allowing for classical communication between subcircuits can improve the sampling overhead. In this work, we derive a closed formula for the optimal sampling overhead for cutting an arbitrary number of two-qubit unitaries and provide the corresponding decomposition. We find that cutting several arbitrary two-qubit unitaries together is cheaper than cutting them individually and classical communication does not give any advantage.

Circuit knitting is the process of partitioning large quantum circuits into smaller subcircuits such that the result of the original circuits can be deduced by only running the subcircuits. Such techniques will be crucial for near-term and early fault-tolerant quantum computers, as the limited number of qubits is likely to be a major bottleneck for demonstrating quantum advantage. One typically distinguishes between gate cuts and wire cuts when partitioning a circuit. The cost for any circuit knitting approach scales exponentially in the number of cuts. One possibility to realize a cut is via the quasiprobability simulation technique. In fact, we argue that all existing rigorous circuit knitting techniques can be understood in this framework. Furthermore, we characterize the optimal overhead for wire cuts where the subcircuits can exchange classical information or not. We show that the optimal cost for cutting n wires without and with classical communication between the subcircuits scales as O(16^n) and O(4^n), respectively.

Circuit cutting is a technique for simulating large quantum circuits by partitioning them into smaller subcircuits, which can be executed on smaller quantum devices. The results from these subcircuits are then combined in classical post-processing to accurately reconstruct the expectation value of the original circuit. Circuit cutting introduces a sampling overhead that grows exponentially with the number of gates and qubit wires that are cut. Many recently developed quasiprobabilistic circuit cutting techniques leverage classical side information, obtained from intermediate measurements within the subcircuits, to enhance the post-processing step. In this work, we provide a formalization of general circuit cutting techniques utilizing side information through quantum instruments. With this framework, we analyze the advantage that classical side information provides in reducing the sampling overhead of circuit cutting. Surprisingly, we find that in certain scenarios, side information does not yield any reduction in sampling overhead, whereas in others it is essential for circuit cutting to be feasible at all. Furthermore, we present a lower bound for the optimal sampling overhead with side information that can be evaluated efficiently via semidefinite programming and improves on all previously known lower bounds.

We introduce a novel capacity measure 2sED for statistical models based on the effective dimension. The new quantity provably bounds the generalization error under mild assumptions on the model. Furthermore, simulations on standard data sets and popular model architectures show that 2sED correlates well with the training error. For Markovian models, we show how to efficiently approximate 2sED from below through a layerwise iterative approach, which allows us to tackle deep learning models with a large number of parameters. Simulation results suggest that the approximation is good for different prominent models and data sets.

The min-entropy of a quantum system A conditioned on another quantum system E describes how much randomness can be extracted from A with respect to an adversary in possession of E. This quantity plays a crucial role in quantum cryptography: the security proofs of many quantum cryptographic protocols reduce to showing a lower bound on such a min-entropy. Here, we develop a new tool, called generalised entropy accumulation, for computing such bounds. Concretely, we consider a sequential process in which each step outputs a system A i and updates a side information register E. We prove that if this process satisfies a natural “non-signalling” condition between past outputs and future side information, the min-entropy of the outputs $A_{1},\ldots,\ A_{n}$ conditioned on the side information E at the end of the process can be bounded from below by a sum of von Neumann entropies associated with the individual steps. This is a generalisation of the entropy accumulation theorem (EAT) [1], which deals with a more restrictive model of side information: there, past side information cannot be updated in subsequent rounds, and newly generated side information has to satisfy a Markov condition.Due to its more general model of side-information, our generalised EAT can be applied more easily and to a broader range of cryptographic protocols. In particular, it is the first general tool that is applicable to mistrustful device-independent cryptography. To demonstrate this, we give the first security proof for blind randomness expansion [2] against general adversaries. Furthermore, our generalised EAT can be used to give improved security proofs for quantum key distribution [3], and also has applications beyond quantum cryptography.

The scarcity of qubits is a major obstacle to the practical usage of quantum computers in the near future. To circumvent this problem, various circuit knitting techniques have been developed to partition large quantum circuits into subcircuits that fit on smaller devices, at the cost of a simulation overhead. In this work, we study a particular method of circuit knitting based on quasiprobability simulation of nonlocal gates with operations that act locally on the subcircuits. We investigate whether classical communication between these local quantum computers can help. We provide a positive answer by showing that for circuits containing $n$ nonlocal CNOT gates connecting two circuit parts, the simulation overhead can be reduced from $O(9^n)$ to $O(4^n)$ if one allows for classical information exchange. Similar improvements can be obtained for general Clifford gates and, at least in a restricted form, for other gates such as controlled rotation gates.

Quantum support vector machines employ quantum circuits to define the kernel function. It has been shown that this approach offers a provable exponential speedup compared to any known classical algorithm for certain data sets. The training of such models corresponds to solving a convex optimization problem either via its primal or dual formulation. Due to the probabilistic nature of quantum mechanics, the training algorithms are affected by statistical uncertainty, which has a major impact on their complexity. We show that the dual problem can be solved in O(M4.67/ε2) quantum circuit evaluations, where M denotes the size of the data set and ε the solution accuracy compared to the ideal result from exact expectation values, which is only obtainable in theory. We prove under an empirically motivated assumption that the kernelized primal problem can alternatively be solved in O(min{M2/ε6,1/ε10}) evaluations by employing a generalization of a known classical algorithm called Pegasos. Accompanying empirical results demonstrate these analytical complexities to be essentially tight. In addition, we investigate a variational approximation to quantum support vector machines and show that their heuristic training achieves considerably better scaling in our experiments.

Making statements about the performance of trained models on tasks involving new data is one of the primary goals of machine learning, i.e., to understand the generalization power of a model. Various capacity measures try to capture this ability, but usually fall short in explaining important characteristics of models that we observe in practice. In this study, we propose the local effective dimension as a capacity measure which seems to correlate well with generalization error on standard data sets. Importantly, we prove that the local effective dimension bounds the generalization error and discuss the aptness of this capacity measure for machine learning models.

Variational quantum time evolution allows us to simulate the time dynamics of quantum systems with near-term compatible quantum circuits. Due to the variational nature of this method the accuracy of the simulation is a priori unknown. We derive global phase agnostic error bounds for the state simulation accuracy with variational quantum time evolution that improve the tightness of fidelity estimates over existing error bounds. These analysis tools are practically crucial for assessing the quality of the simulation and making informed choices about simulation hyper-parameters. The efficient, a posteriori evaluation of the bounds can be tightly integrated with the variational time simulation and, hence, results in a minor resource overhead which is governed by the system's energy variance. The performance of the novel error bounds is demonstrated on numerical examples.

Brascamp-Lieb inequalities are entropy inequalities which have a dual formulation as generalized Young inequalities. In this work, we introduce a fully quantum version of this duality, relating quantum relative entropy inequalities to matrix exponential inequalities of Young type. We demonstrate this novel duality by means of examples from quantum information theory-including entropic uncertainty relations, strong data-processing inequalities, super-additivity inequalities, and many more. As an application we find novel uncertainty relations for Gaussian quantum operations that can be interpreted as quantum duals of the well-known family of 'geometric' Brascamp-Lieb inequalities.

Quantum support vector machines have the potential to achieve a quantum speedup for solving certain machine learning problems. The key challenge for doing so is finding good quantum kernels for a given data set -- a task called kernel alignment. In this paper we study this problem using the Pegasos algorithm, which is an algorithm that uses stochastic gradient descent to solve the support vector machine optimization problem. We extend Pegasos to the quantum case and and demonstrate its effectiveness for kernel alignment. Unlike previous work which performs kernel alignment by training a QSVM within an outer optimization loop, we show that using Pegasos it is possible to simultaneously train the support vector machine and align the kernel. Our experiments show that this approach is capable of aligning quantum feature maps with high accuracy, and outperforms existing quantum kernel alignment techniques. Specifically, we demonstrate that Pegasos is particularly effective for non-stationary data, which is an important challenge in real-world applications.

Quantum computations are typically performed as a sequence of basic operations, called quantum gates. Different gate sequences, called quantum circuits, can implement the same overall quantum computation. Since every additional quantum gate takes time and introduces noise into the system, it is important to find the smallest possible quantum circuit that implements a given computation, especially for near-term quantum devices that can execute only a limited number of quantum gates before noise renders the computation useless. An important building block for many quantum circuit optimization techniques is pattern matching: given a large and small quantum circuit, we would like to find all maximal matches of the small circuit, called a pattern , in the large circuit, considering pairwise commutation of quantum gates. In this work, we present the first classical algorithm for pattern matching that provably finds all maximal matches and is efficient enough to be practical for circuit sizes typical for near-term devices. We demonstrate numerically 1 that combining our algorithm with known pattern-matching-based circuit optimization techniques reduces the gate count of a random quantum circuit by ∼ 30% and can further improve practically relevant quantum circuits that were already optimized with state-of-the-art techniques.

Quantum error mitigation techniques can reduce noise on current quantum hardware without the need for fault-tolerant quantum error correction. For instance, the quasiprobability method simulates a noise-free quantum computer using a noisy one, with the caveat of only producing the correct expected values of observables. The cost of this error mitigation technique manifests as a sampling overhead which scales exponentially in the number of corrected gates. In this work, we present a new algorithm based on mathematical optimization that aims to choose the quasiprobability decomposition in a noise-aware manner. This directly leads to a significantly lower basis of the sampling overhead compared to existing approaches. A key element of the novel algorithm is a robust quasiprobability method that allows for a tradeoff between an approximation error and the sampling overhead via semidefinite programming.