Diamond Norm Research Articles

High-Order Randomized Compiler for Hamiltonian Simulation

Hamiltonian simulation is known to be one of the fundamental building blocks of a variety of quantum algorithms such as its most immediate application, that of simulating many-body systems to extract their physical properties. In this work, we present qSWIFT, a high-order randomized algorithm for Hamiltonian simulation. In qSWIFT, the required number of gates for a given precision is independent of the number of terms in the Hamiltonian, while the systematic error is exponentially reduced with regard to the order parameter. In this respect, our qSWIFT is a higher-order counterpart of the previously proposed quantum stochastic drift protocol (qDRIFT), the number of gates in which scales linearly with the inverse of the precision required. We construct the qSWIFT channel and establish a rigorous bound for the systematic error quantified by the diamond norm. qSWIFT provides an algorithm to estimate given physical quantities by using a system with one ancilla qubit, which is as simple as other product-formula-based approaches such as regular Trotter-Suzuki decompositions and qDRIFT. Our numerical experiment reveals that the required number of gates in qSWIFT is significantly reduced compared to qDRIFT. In particular, the advantage is significant for problems where high precision is required; e.g., to achieve a systematic relative propagation error of 10−6, the required number of gates in third-order qSWIFT is 1000 times smaller than that of qDRIFT. Published by the American Physical Society 2024

Error bounds for Lie group representations in quantum mechanics

We provide state-dependent error bounds for strongly continuous unitary representations of connected Lie groups. That is, we bound the difference of two unitaries applied to a state in terms of the energy with respect to a reference Hamiltonian associated with the representation and a left-invariant metric distance on the group. Our method works for any connected Lie group, and the metric is independent of the chosen representation. The approach also applies to projective representations and allows us to provide bounds on the energy-constrained diamond norm distance of any suitably continuous channel representation of the group.

Shorter quantum circuits via single-qubit gate approximation

We give a novel procedure for approximating general single-qubit unitaries from a finite universal gate set by reducing the problem to a novel magnitude approximation problem, achieving an immediate improvement in sequence length by a factor of 7/9. Extending the works \cite{Hastings2017} and \cite{Campbell2017}, we show that taking probabilistic mixtures of channels to solve fallback \cite{BRS2015} and magnitude approximation problems saves factor of two in approximation costs. In particular, over the Clifford+T gate set we achieve an average non-Clifford gate count of 0.23log2⁡(1/ε)+2.13 and T-count 0.56log2⁡(1/ε)+5.3 with mixed fallback approximations for diamond norm accuracy ε.This paper provides a holistic overview of gate approximation, in addition to these new insights. We give an end-to-end procedure for gate approximation for general gate sets related to some quaternion algebras, providing pedagogical examples using common fault-tolerant gate sets (V, Clifford+T and Clifford+T). We also provide detailed numerical results for Clifford+T and Clifford+T gate sets. In an effort to keep the paper self-contained, we include an overview of the relevant algorithms for integer point enumeration and relative norm equation solving. We provide a number of further applications of the magnitude approximation problems, as well as improved algorithms for exact synthesis, in the Appendices.

Benchmarking quantum logic operations relative to thresholds for fault tolerance

Contemporary methods for benchmarking noisy quantum processors typically measure average error rates or process infidelities. However, thresholds for fault-tolerant quantum error correction are given in terms of worst-case error rates—defined via the diamond norm—which can differ from average error rates by orders of magnitude. One method for resolving this discrepancy is to randomize the physical implementation of quantum gates, using techniques like randomized compiling (RC). In this work, we use gate set tomography to perform precision characterization of a set of two-qubit logic gates to study RC on a superconducting quantum processor. We find that, under RC, gate errors are accurately described by a stochastic Pauli noise model without coherent errors, and that spatially correlated coherent errors and non-Markovian errors are strongly suppressed. We further show that the average and worst-case error rates are equal for randomly compiled gates, and measure a maximum worst-case error of 0.0197(3) for our gate set. Our results show that randomized benchmarks are a viable route to both verifying that a quantum processor’s error rates are below a fault-tolerance threshold, and to bounding the failure rates of near-term algorithms, if—and only if—gates are implemented via randomization methods which tailor noise.

Operational Quantum Average-Case Distances

We introduce distance measures between quantum states, measurements, and channels based on their statistical distinguishability in generic experiments. Specifically, we analyze the average Total Variation Distance (TVD) between output statistics of protocols in which quantum objects are intertwined with random circuits and measured in standard basis. We show that for circuits forming approximate 4-designs, the average TVDs can be approximated by simple explicit functions of the underlying objects – the average-case distances (ACDs). We apply them to analyze the effects of noise in quantum advantage experiments and for efficient discrimination of high-dimensional states and channels without quantum memory. We argue that ACDs are better suited for assessing the quality of NISQ devices than common distance measures such as trace distance or the diamond norm.

Linear growth of circuit complexity from Brownian dynamics

How rapidly can a many-body quantum system generate randomness? Using path integral methods, we demonstrate that Brownian quantum systems have circuit complexity that grows linearly with time. In particular, we study Brownian clusters of N spins or fermions with time-dependent all-to-all interactions, and calculate the Frame Potential to characterize complexity growth in these models. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using path integral methods. Within this framework it is straightforward to show that the kth Frame Potential comes within ϵ of the Haar value after a time of order t ~ kN + k log k + log ϵ−1. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary k-design after a time of order t ~ kN. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a k-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are analytically tractable and are directly applicable to practical applications calling for unitary k-designs.

Exploring Quantum Average-Case Distances: Proofs, Properties, and Examples

In this work, we present an in-depth study of average-case quantum distances introduced in [1]. The average-case distances approximate, up to the relative error, the average Total-Variation (TV) distance between measurement outputs of two quantum processes, in which quantum objects of interest (states, measurements, or channels) are intertwined with random quantum circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantify <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">average-case</i> statistical distinguishability via random quantum circuits. We prove that once a family of random circuits forms an δ-approximate 4-design, with δ = <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">–8</sup> ), then the average-case distances can be approximated by simple explicit functions that can be expressed via simple degree two polynomials in objects of interest. For systems of moderate dimension, they can be easily explicitly computed – no optimization is needed as opposed to diamond norm distance between channels or operational distance between measurements. We prove that those functions, which we call quantum average-case distances, have a plethora of desirable properties, such as subadditivity w.r.t. tensor products, joint convexity, and (restricted) data-processing inequalities. Notably, all distances utilize the Hilbert-Schmidt (HS) norm, which provides this norm with a new operational interpretation. We also provide upper bounds on the maximal ratio between worst-case and average-case distances, and for each of them, we provide an example that saturates the bound. Specifically, we show that for each dimension <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> this ratio is at most <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> , <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/2</sup> for states, measurements, and channels, respectively. To support the practical usefulness of our findings, we study multiple examples in which average-case quantum distances can be calculated analytically.

Distance-based resource quantification for sets of quantum measurements

The advantage that quantum systems provide for certain quantum information processing tasks over their classical counterparts can be quantified within the general framework of resource theories. Certain distance functions between quantum states have successfully been used to quantify resources like entanglement and coherence. Perhaps surprisingly, such a distance-based approach has not been adopted to study resources of quantum measurements, where other geometric quantifiers are used instead. Here, we define distance functions between sets of quantum measurements and show that they naturally induce resource monotones for convex resource theories of measurements. By focusing on a distance based on the diamond norm, we establish a hierarchy of measurement resources and derive analytical bounds on the incompatibility of any set of measurements. We show that these bounds are tight for certain projective measurements based on mutually unbiased bases and identify scenarios where different measurement resources attain the same value when quantified by our resource monotone. Our results provide a general framework to compare distance-based resources for sets of measurements and allow us to obtain limitations on Bell-type experiments.

Randomized Benchmarking beyond Groups

Randomized benchmarking (RB) is the gold standard for experimentally evaluating the quality of quantum operations. The current framework for RB is centered on groups and their representations, but this can be problematic. For example, Clifford circuits need up to $O(n^2)$ gates, and thus Clifford RB cannot scale to larger devices. Attempts to remedy this include new schemes such as linear cross-entropy benchmarking (XEB), cycle benchmarking, and non-uniform RB, but they do not fall within the group-based RB framework. In this work, we formulate the \emph{universal randomized benchmarking (URB) framework} which does away with the group structure and also replaces the recovery gate plus measurement component with a general ``post-processing'' POVM. Not only does this framework cover most of the existing benchmarking schemes, but it also gives the language for and helps inspire the formulation of new schemes. We specifically consider a class of URB schemes called \emph{twirling schemes}. For twirling schemes, the post-processing POVM approximately factorizes into an intermediate channel, inverting maps, and a final measurement. This leads us to study the twirling map corresponding to the gate ensemble specified by the scheme. We prove that if this twirling map is strictly within unit distance of the Haar twirling map in induced diamond norm, the probability of measurement as a function of gate length is a single exponential decay up to small error terms. The core technical tool we use is the matrix perturbation theory of linear operators on quantum channels.

Divide-and-conquer verification method for noisy intermediate-scale quantum computation

Several noisy intermediate-scale quantum computations can be regarded as logarithmic-depth quantum circuits on a sparse quantum computing chip, where two-qubit gates can be directly applied on only some pairs of qubits. In this paper, we propose a method to efficiently verify such noisy intermediate-scale quantum computation. To this end, we first characterize small-scale quantum operations with respect to the diamond norm. Then by using these characterized quantum operations, we estimate the fidelity &amp;#x27E8;&amp;#x03C8;t|&amp;#x03C1;&amp;#x005E;out|&amp;#x03C8;t&amp;#x27E9; between an actual n-qubit output state &amp;#x03C1;&amp;#x005E;out obtained from the noisy intermediate-scale quantum computation and the ideal output state (i.e., the target state) |&amp;#x03C8;t&amp;#x27E9;. Although the direct fidelity estimation method requires O(2n) copies of &amp;#x03C1;&amp;#x005E;out on average, our method requires only O(D3212D) copies even in the worst case, where D is the denseness of |&amp;#x03C8;t&amp;#x27E9;. For logarithmic-depth quantum circuits on a sparse chip, D is at most O(log&amp;#x2061;n), and thus O(D3212D) is a polynomial in n. By using the IBM Manila 5-qubit chip, we also perform a proof-of-principle experiment to observe the practical performance of our method.

On the connection between quantum pseudorandomness and quantum hardware assumptions

This paper, for the first time, addresses the questions related to the connections between quantum pseudorandomness and quantum hardware assumptions, specifically quantum physical unclonable functions (qPUFs). Our results show that efficient pseudorandom quantum states (PRS) are sufficient to construct the challenge set for universally unforgeable qPUFs, improving the previous existing constructions based on the Haar-random states. We also show that both the qPUFs and the quantum pseudorandom unitaries (PRUs) can be constructed from each other, providing new ways to obtain PRS from the hardware assumptions. Moreover, we provide a sufficient condition (in terms of the diamond norm) that a set of unitaries should have to be a PRU in order to construct a universally unforgeable qPUF, giving yet another novel insight into the properties of the PRUs. Later, as an application of our results, we show that the efficiency of an existing qPUF-based client–server identification protocol can be improved without losing the security requirements of the protocol.

Two-way unclonable encryption with a vulnerable sender

Unclonable Encryption, introduced by Gottesman in 2003 [Quantum Inform. Comput. 3 (2003) 581], is a quantum protocol that guarantees the secrecy of a successfully transferred classical message even when all keys leak at a later time. We propose an Unclonable Encryption protocol with the additional property that the sender’s key material is allowed to leak even in the case of an unsuccessful run. This extra feature makes it possible to achieve secure quantum encryption even when one of the parties is unable to protect its keys against after-protocol theft. Such an asymmetry occurs e.g. in case of server–client scenarios, where the client device is resource constrained and/or located in a hostile environment. Our protocol makes use of a bidirectional quantum channel in a manner similar to the two-way protocol LM05 [Phys. Rev. Lett. 94 (2005) 140501]. Bob sends random qubit states to Alice. Alice flips the states in a way that depends on the message and a shared key, and sends the resulting states back to Bob. Bob recovers Alice’s message by measuring the flips. We prove that our protocol satisfies the definition of unclonable encryption and additionally that the message remains secure even if all of Alice’s keys leak after the protocol. Furthermore, we show that some of the key materials can be safely re-used. Our security proof is formulated in terms of diamond norms, which makes it composable, and allows for noisy quantum channels. We work out the details only for the asymptotics in the limit of long messages. As a side result, we construct a two-way QKD scheme with a high key rate. We show that its key rate is higher than the rate of the two-way QKD scheme LM05 proven by Beaudry et al. [Phys. Rev. A 88 (2013) 062302] for the case of independent channel noise.

Qubit-based unclonable encryption with key recycling

We re-visit Unclonable Encryption as introduced by Gottesman in~2003 \cite{uncl}. We look at the combination of Unclonable Encryption and Key Recycling, while aiming for low communication complexity and high rate. We introduce a qubit-based prepare-and-measure Unclonable Encryption scheme with re-usable keys. Our scheme consists of a single transmission by Alice and a single classical feedback from Bob. The transmission from Alice to Bob consists entirely of qubits. The rate, defined as the message length divided by the number of qubits, is higher than what can be achieved using Gottesman's scheme~\cite{uncl}. We provide a security proof based on the diamond norm distance, taking noise into account.

Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps

Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels — that is, not completely positive — can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.

Time-varying quantum channel models for superconducting qubits

The decoherence effects experienced by the qubits of a quantum processor are generally characterized using the amplitude damping time (T1) and the dephasing time (T2). Quantum channel models that exist at the time of writing assume that these parameters are fixed and invariant. However, recent experimental studies have shown that they exhibit a time-varying (TV) behaviour. These time-dependant fluctuations of T1 and T2, which become even more pronounced in the case of superconducting qubits, imply that conventional static quantum channel models do not capture the noise dynamics experienced by realistic qubits with sufficient precision. In this article, we study how the fluctuations of T1 and T2 can be included in quantum channel models. We propose the idea of time-varying quantum channel (TVQC) models, and we show how they provide a more realistic portrayal of decoherence effects than static models in some instances. We also discuss the divergence that exists between TVQCs and their static counterparts by means of a metric known as the diamond norm. In many circumstances this divergence can be significant, which indicates that the time-dependent nature of decoherence must be considered, in order to construct models that capture the real nature of quantum devices.

Energy-Constrained Discrimination of Unitaries, Quantum Speed Limits, and a Gaussian Solovay-Kitaev Theorem.

We investigate the energy-constrained (EC) diamond norm distance between unitary channels acting on possibly infinite-dimensional quantum systems, and establish a number of results. First, we prove that optimal EC discrimination between two unitary channels does not require the use of any entanglement. Extending a result by Acín, we also show that a finite number of parallel queries suffices to achieve zero error discrimination even in this EC setting. Second, we employ EC diamond norms to study a novel type of quantum speed limits, which apply to pairs of quantum dynamical semigroups. We expect these results to be relevant for benchmarking internal dynamics of quantum devices. Third, we establish a version of the Solovay-Kitaev theorem that applies to the group of Gaussian unitaries over a finite number of modes, with the approximation error being measured with respect to the EC diamond norm relative to the photon number Hamiltonian.

Experimental Pauli-frame randomization on a superconducting qubit

The promise of quantum computing with imperfect qubits relies on the ability of a quantum computing system to scale cheaply through error correction and fault-tolerance. While fault-tolerance requires relatively mild assumptions about the nature of qubit errors, the overhead associated with coherent and non-Markovian errors can be orders of magnitude larger than the overhead associated with purely stochastic Markovian errors. One proposal to address this challenge is to randomize the circuits of interest, shaping the errors to be stochastic Pauli errors but leaving the aggregate computation unaffected. The randomization technique can also suppress couplings to slow degrees of freedom associated with non-Markovian evolution. Here we demonstrate the implementation of Pauli-frame randomization in a superconducting circuit system, exploiting a flexible programming and control infrastructure to achieve this with low effort. We use high-accuracy gate-set tomography to characterize in detail the properties of the circuit error, with and without the randomization procedure, which allows us to make rigorous statements about Markovianity as well as the nature of the observed errors. We demonstrate that randomization suppresses signatures of non-Markovian evolution to statistically insignificant levels, from a Markovian model violation ranging from $43\sigma$ to $1987\sigma$, down to violations between $0.3\sigma$ and $2.7\sigma$ under randomization. Moreover, we demonstrate that, under randomization, the experimental errors are well described by a Pauli error model, with model violations that are similarly insignificant (between $0.8\sigma$ and $2.7\sigma$). Importantly, all these improvements in the model accuracy were obtained without degradation to fidelity, and with some improvements to error rates as quantified by the diamond norm.

Bounds on amplitude-damping-channel discrimination

Amplitude-damping (AD) channels are good models for many physical scenarios, and so the development of protocols to discriminate between them is an important task in quantum information science. It is therefore important to bound the performance of such protocols. Since adaptivity has been shown to improve the performance of discrimination protocols, bounds on the distinguishability of AD channels must take this into account. In this paper, we use both channel simulation and a bound based on the diamond norm to significantly tighten the upper bound on the trace norm between the possible outputs of binary channel discrimination protocols acting on AD channels (and hence the lower bound on the error probability of such protocols). The diamond norm between any two AD channels is found analytically, giving the optimal error probability for a one-shot discrimination protocol. We also present a tighter lower bound on the achievable trace norm between protocol outputs (and a corresponding upper bound on the achievable error probability). The upper and lower bounds are compared with existing bounds and then applied to quantum hacking and biological quantum sensing scenarios.

Convergence Rates for the Quantum Central Limit Theorem

Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state rho with finite second moments, converges to the Gaussian state with the same first and second moments as rho . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate mathcal {O}left( n^{-1/2}right) in the Hilbert–Schmidt norm whenever the third moments of rho are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of n beam splitters of equal transmissivities lambda ^{1/n} fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate mathcal {O}Big (n^{-frac{1}{2(m+1)}}Big ). This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function chi _rho is uniformly bounded by some eta _rho <1 outside of any neighbourhood of the origin; also, eta _rho can be made to depend only on the energy of the state rho .

Refined diamond norm bounds on the emergence of objectivity of observables

The theory of quantum Darwinism aims to explain how our objective classical reality arises from the quantum world, by analysing the distribution of information about a quantum system that is accessible to multiple observers, who probe the system by intercepting fragments of its environment. Previous work showed that, when the number of environmental fragments grows, the quantum channels modelling the information flow from system to observers become arbitrarily close—in terms of diamond norm distance—to ‘measure-and-prepare’ channels, ensuring objectivity of observables; the convergence is formalised by an upper bound on the diamond norm distance, which decreases with increasing number of fragments. Here, we derive tighter diamond norm bounds on the emergence of objectivity of observables for quantum systems of infinite dimension, providing an approach which can bridge between the finite- and the infinite-dimensional cases. Furthermore, we probe the tightness of our bounds by considering a specific model of a system-environment dynamics given by a pure loss channel. Finally, we generalise to infinite dimensions a result obtained by Brandão et al (2015 Nat. Commun. 6 7908), which provides an operational characterisation of quantum discord in terms of one-sided redistribution of correlations to many parties. Our results provide a unifying framework to benchmark quantitatively the rise of objectivity in the quantum-to-classical transition.