On the moments of random quantum circuits and robust quantum complexity

We prove new lower bounds on the growth of robust quantum circuit complexity -- the minimal number of gates $C_{\delta}(U)$ to approximate a unitary $U$ up to an error of $\delta$ in operator norm distance. More precisely we show two bounds for random quantum circuits with local gates drawn from a subgroup of $SU(4)$. First, for $\delta=\Theta(2^{-n})$, we prove a linear growth rate: $C_{\delta}\geq d/\mathrm{poly}(n)$ for random quantum circuits on $n$ qubits with $d\leq 2^{n/2}$ gates. Second, for $\delta=\Omega(1)$, we prove a square-root growth of complexity: $C_{\delta}\geq \sqrt{d}/\mathrm{poly}(n)$ for all $d\leq 2^{n/2}$. Finally, we provide a simple conjecture regarding the Fourier support of randomly drawn Boolean functions that would imply linear growth for constant $\delta$. While these results follow from bounds on the moments of random quantum circuits, we do not make use of existing results on the generation of unitary $t$-designs. Instead, we bound the moments of an auxiliary random walk on the diagonal unitaries acting on phase states. In particular, our proof is comparably short and self-contained.Comment: 13 pages, 1 figure, v2: modified main theorem due to a gap in v

Emergent statistical mechanics from properties of disordered random matrix product states

The study of generic properties of quantum states has led to an abundance of insightful results. A meaningful set of states that can be efficiently prepared in experiments are ground states of gapped local Hamiltonians, which are well approximated by matrix product states. In this work, we introduce a picture of generic states within the trivial phase of matter with respect to their non-equilibrium and entropic properties: We do so by rigorously exploring non-translation-invariant matrix product states drawn from a local i.i.d. Haar-measure. We arrive at these results by exploiting techniques for computing moments of random unitary matrices and by exploiting a mapping to partition functions of classical statistical models, a method that has lead to valuable insights on local random quantum circuits. Specifically, we prove that such disordered random matrix product states equilibrate exponentially well with overwhelming probability under the time evolution of Hamiltonians featuring a non-degenerate spectrum. Moreover, we prove two results about the entanglement Renyi entropy: The entropy with respect to sufficiently disconnected subsystems is generically extensive in the system-size, and for small connected systems the entropy is almost maximal for sufficiently large bond dimensions.Comment: 11 page

Shallow shadows: Expectation estimation using low-depth random Clifford circuits

We provide practical and powerful schemes for learning many properties of an unknown n-qubit quantum state using a sparing number of copies of the state. Specifically, we present a depth-modulated randomized measurement scheme that interpolates between two known classical shadows schemes based on random Pauli measurements and random Clifford measurements. These can be seen within our scheme as the special cases of zero and infinite depth, respectively. We focus on the regime where depth scales logarithmically in n and provide evidence that this retains the desirable properties of both extremal schemes whilst, in contrast to the random Clifford scheme, also being experimentally feasible. We present methods for two key tasks; estimating expectation values of certain observables from generated classical shadows and, computing upper bounds on the depth-modulated shadow norm, thus providing rigorous guarantees on the accuracy of the output estimates. We consider observables that can be written as a linear combination of poly(n) Paulis and observables that can be written as a low bond dimension matrix product operator. For the former class of observables both tasks are solved efficiently in n. For the latter class, we do not guarantee efficiency but present a method that works in practice; by variationally computing a heralded approximate inverses of a tensor network that can then be used for efficiently executing both these tasks.Comment: 22 pages, 12 figures. Version 2: new MPS variational inversion algorithm and new numeric

Efficient Unitary Designs with a System-Size Independent Number of Non-Clifford Gates

Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full n-qubit group, one often resorts to t-designs. Unitary t-designs mimic the Haar-measure up to t-th moments. It is known that Clifford operations can implement at most 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject O(t4log2(t)log(1/ε)) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an ε-approximate t-design. Strikingly, the number of non-Clifford gates required is independent of the system size – asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the t-th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators

Dynamical structure factors of dynamical quantum simulators

The dynamical structure factor is one of the experimental quantities crucial in scrutinizing the validity of the microscopic description of strongly correlated systems. However, despite its long-standing importance, it is exceedingly difficult in generic cases to numerically calculate it, ensuring that the necessary approximations involved yield a correct result. Acknowledging this practical difficulty, we discuss in what way results on the hardness of classically tracking time evolution under local Hamiltonians are precisely inherited by dynamical structure factors and, hence, offer in the same way the potential computational capabilities that dynamical quantum simulators do: We argue that practically accessible variants of the dynamical structure factors are bounded-error quantum polynomial time (BQP)-hard for general local Hamiltonians. Complementing these conceptual insights, we improve upon a novel, readily available measurement setup allowing for the determination of the dynamical structure factor in different architectures, including arrays of ultra-cold atoms, trapped ions, Rydberg atoms, and superconducting qubits. Our results suggest that quantum simulations employing near-term noisy intermediate-scale quantum devices should allow for the observation of features of dynamical structure factors of correlated quantum matter in the presence of experimental imperfections, for larger system sizes than what is achievable by classical simulation.European Research Council (Taming Non-Equilibrium Quantum Systems)Templeton FoundationFoundational Questions InstituteGerman Research Foundation (DFG) EI 519/14-1 EI 519/15-1 CRC 183 FOR 2724MATH+European Union Horizon 2020 research and innovation program 817482European Union (EU) 754446University of Granada Research and Knowledge Transfer Fund-Athenea3

Resource theory of quantum uncomplexity

Quantum complexity is emerging as a key property of many-body systems, including black holes, topological materials, and early quantum computers. A state's complexity quantifies the number of computational gates required to prepare the state from a simple tensor product. The greater a state's distance from maximal complexity, or “uncomplexity,” the more useful the state is as input to a quantum computation. Separately, resource theories—simple models for agents subject to constraints—are burgeoning in quantum information theory. We unite the two domains, confirming Brown and Susskind's conjecture that a resource theory of uncomplexity can be defined. The allowed operations, fuzzy operations, are slightly random implementations of two-qubit gates chosen by an agent. We formalize two operational tasks, uncomplexity extraction and expenditure. Their optimal efficiencies depend on an entropy that we engineer to reflect complexity. We also present two monotones, uncomplexity measures that decline monotonically under fuzzy operations, in certain regimes. This work unleashes on many-body complexity the resource-theory toolkit from quantum information theory

Improved spectral gaps for random quantum circuits: Large local dimensions and all-to-all interactions

Random quantum circuits are a central concept in quantum information theory with applications ranging from demonstrations of quantum computational advantage to descriptions of scrambling in strongly interacting systems and black holes. The utility of random quantum circuits in these settings stems from their ability to rapidly generate quantum pseudorandomness. In a seminal paper by Brandão, Harrow, and Horodecki [Commun. Math. Phys. 346, 397 (2016)] it was proven that the tth moment operator of local random quantum circuits on n qudits with local dimension q has a spectral gap of at least Ω(n−1t−5−3.1/ln(q)), which implies that they are efficient constructions of approximate unitary designs. As a first result, we use Knabe bounds for the spectral gaps of frustration-free Hamiltonians to show that one-dimensional random quantum circuits have a spectral gap scaling as Ω(n−1), provided that t is small compared to the local dimension: t2≤O(q). This implies a (nearly) linear scaling of the circuit depth in the design order t. Our second result is an unconditional spectral gap bounded below by Ω[n−1ln−1(n)t−α(q)] for random quantum circuits with all-to-all interactions. This improves both the n and t scaling in design depth for the nonlocal model. We show this by proving a recursion relation for the spectral gaps involving an auxiliary random walk. Lastly, we solve the smallest nontrivial case exactly and combine with numerics and Knabe bounds to improve the constants involved in the spectral gap for small values of t

Zufall und Komplexität in zufälligen komplexen Quantensystemen

The interaction of computational complexity and quantum physics touches a wide range of topics from emerging technologies such as quantum computers to the physics of black holes. While tools from quantum information theory can help to answer questions in theoretical computer science, conversely, the ideas developed for analyzing the power of classical computers can shed light on physical phenomena. Deeply intertwined with both, quantum theory and the theory of complexity, is randomness. Indeed, quantum theory is a probabilistic theory and can predict, in general, only expectation values for observables. In the theory of algorithms, randomness is not only a key design primitive but also indispensable as a proof technique. In this thesis we make advances at the intersection of randomness, complexity and quantum theory. This includes a mathematical analysis of random ensembles of tensor network states, leading to new results on the average-case complexity of tensor network contraction, contributions to the foundation of verifiable quantum supremacy experiments as well as novel bounds on the generation of quantum pseudorandomness. In particular, we show that unitary t-designs can be generated with a system-size independent number of non-Clifford resources and that random quantum circuits generate designs in depth O(nt^(5+o(1))). These results have numerous applications including the best bounds on the growth of operational notions of quantum circuit complexity. Moreover, we provide a proof of the Brown-Susskind conjecture for the linear growth of exact circuit complexity in random quantum circuits. The majority of the results in this thesis are theorems published in academic journals. The tools exploited for this analysis range from the concepts of theoretical computer science, such as complexity classes, over ideas from harmonic analysis and Markov chains to the techniques of quantum many-body physics. In an appendix, this thesis contains several unpublished results such as the first non-trivial upper bounds on moments of the permanent and generation of quantum pseudorandomness with random measurements on the cluster state.Die Schnittstelle von Komplexitätstheorie und Quantenphysik umfasst Quantentechnologien bis hin zu fundamentalen Fragen über die Physik schwarzer Löcher. Während die Methoden der Quanteninformationstheorie dabei helfen können, Fragen in der Informatik zu beantworten, tragen algorithmische Ideen dazu bei physikalische Phänomene zu erklären. Sowohl in der Komplexitätstheorie als auch in der Quantentheorie ist das Konzept des Zufalls allgemeingegenwärtig. Für die Entwicklung neuer Algorithmen ist Zufall ein mächtiges Werkzeug und erlaubt oft effiziente Methoden, wo es keine schnellen deterministischen Lösungen gibt. Die Quantentheorie ist inhärent probabilistisch und erlaubt nur Vorhersagen über Erwartungswerte. In dieser Dissertation machen wir mehrere Fortschritte an der mathematischen Theorie dieser Schnittstelle. Das beinhaltet die Untersuchung von zufälligen Ensembles sogenannter Tensorproduktzustände, Komplexitätsresultate für den typischen Fall von Tensornetzwerken, rigorose Evidenz für verifizierbare Quantenüberlegenheitssexperimente und mehrere neue Schranken auf die Erzeugung von Quantenpseudozufall. Insbesondere zeigen wir, dass unitäre t-Designs mit einer systemunabhängigen Anzahl an nicht-Clifford Gattern erzeugt werden können und das zufällige Quantenschaltkreise t-Designs in einer Tiefe von O(nt^(5+o(1))) erzeugen. Diese Resultate haben zahlreiche Anwendungen und implizieren insbesondere die momentan besten Schranken auf das Wachstum fehlerrobuster Definitionen von Schaltkreiskomplexität. Letztlich enthält diese Dissertation einen Beweis der Brown-Susskind Vermutung für das lineare Wachstum der exakten Schaltkreiskomplexität in zufälligen Quantenschaltkreisen. Die Mehrzahl der Ergebnisse in dieser Arbeit sind mathematische Theoreme mit rigorosen Beweisen. Die Methoden für diese Analyse rangieren von den Konzepten der theoretischen Informatik, über harmonische Analysis und Markovketten zu den Techniken der Vielteilchentheorie. In einem Appendix enthält diese Arbeit mehrere unveröffentlichte Resultate wie die ersten nicht-trivialen oberen Schranken auf Momente der Permanente zufälliger Matrizen so wie die Erzeugung von Pseudozufall mit zufälligen Messungen auf Cluster-Zuständen