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The `quantum gravity in the lab' paradigm suggests that quantum computers might shed light on quantum gravity by simulating the CFT side of the AdS/CFT correspondence and mapping the results to the AdS side. This relies on the assumption that the duality map (the `dictionary') is efficient to compute. In this work, we show that the complexity of the AdS/CFT dictionary is surprisingly subtle: there might be cases in which one can efficiently apply operators to the CFT state (a task we call 'operator reconstruction') without being able to extract basic properties of the dual bulk state such as its geometry (which we call 'geometry reconstruction'). Geometry reconstruction corresponds to the setting where we want to extract properties of a completely unknown bulk dual from a simulated CFT boundary state. We demonstrate that geometry reconstruction may be generically hard due to the connection between geometry and entanglement in holography. In particular we construct ensembles of states whose entanglement approximately obey the Ryu-Takayanagi formula for arbitrary geometries, but which are nevertheless computationally indistinguishable. This suggests that even for states with the special entanglement structure of holographic CFT states, geometry reconstruction might be hard. This result should be compared with existing evidence that operator reconstruction is generically easy in AdS/CFT. A useful analogy for the difference between these two tasks is quantum fully homomorphic encryption (FHE): this encrypts quantum states in such a way that no efficient adversary can learn properties of the state, but operators can be applied efficiently to the encrypted state. We show that quantum FHE can separate the complexity of geometry reconstruction vs operator reconstruction, which raises the question whether FHE could be a useful lens through which to view AdS/CFT., Comment: 45 pages, 9 figures

We investigate the link between quantum position-verification (QPV) and holography established in [MPS19] using holographic quantum error correcting codes as toy models. By inserting the "temporal" scaling of the AdS metric by hand via the bulk Hamiltonian interaction strength, we recover a toy model with consistent causality structure. This leads to an interesting implication between two topics in quantum information: if position-based verification is secure against attacks with small entanglement then there are new fundamental lower bounds for resources required for one Hamiltonian to simulate another., Comment: (v2: updated presentation)

We study the complexity of a classic problem in computational topology, the homology problem: given a description of some space $X$ and an integer $k$, decide if $X$ contains a $k$-dimensional hole. The setting and statement of the homology problem are completely classical, yet we find that the complexity is characterized by quantum complexity classes. Our result can be seen as an aspect of a connection between homology and supersymmetric quantum mechanics. We consider clique complexes, motivated by the practical application of topological data analysis (TDA). The clique complex of a graph is the simplicial complex formed by declaring every $k+1$-clique in the graph to be a $k$-simplex. Our main result is that deciding whether the clique complex of a weighted graph has a hole or not, given a suitable promise on the gap, is $\text{QMA}_1$-hard and contained in $\text{QMA}$. Our main innovation is a technique to lower bound the eigenvalues of the combinatorial Laplacian operator. For this, we invoke a tool from algebraic topology known as \emph{spectral sequences}. In particular, we exploit a connection between spectral sequences and Hodge theory. Spectral sequences will play a role analogous to perturbation theory for combinatorial Laplacians. In addition, we develop the simplicial surgery technique used in prior work. Our result provides some suggestion that the quantum TDA algorithm \emph{cannot} be dequantized. More broadly, we hope that our results will open up new possibilities for quantum advantage in topological data analysis.

Understanding how to characterise and mitigate errors is a key challenge in developing reliable quantum architecture for near-term applications. Recent work (arXiv:2103.17243) provides an efficient set of algorithms for analysing unknown noise processes requiring only tomographic snapshots of the quantum operator under consideration, without the need of any a-priori information on the noise model, nor necessitating a particular experimental setup. The only assumption made is that the observed channel can be approximated by a time-independent Markovian map, which is a typically reasonable framework when considering short time scales. In this note we lift the time-independent assumption, presenting an extension of the scheme now able to analyse noisy dynamics with time-dependent generators from a sequence of snapshots. We hence provide a diagnostic tool for a wider spectrum of instances while inheriting all the favourable features from the previous protocol. On the theoretical side, the problem of characterising time-dependent Markovian channels has been an open problem for many decades. This work gives an approach to tackle this characterisation problem rigorously., Comment: 15 pages

The AdS/CFT correspondence realises the holographic principle where information in the bulk of a space is encoded at its border. We are yet a long way from a full mathematical construction of AdS/CFT, but toy models in the form of holographic quantum error correcting codes (HQECC) have replicated some interesting features of the correspondence. In this work we construct new HQECCs built from random stabilizer tensors that describe a duality between models encompassing local Hamiltonians whilst exactly obeying the Ryu-Takayanagi entropy formula for all boundary regions. We also obtain complementary recovery of local bulk operators for any boundary bipartition. Existing HQECCs have been shown to exhibit these properties individually, whereas our mathematically rigorous toy models capture these features of AdS/CFT simultaneously, advancing further towards a complete construction of holographic duality., Comment: 42 pages, 4 figures. v4: updated to published version, expanded introduction and new figure demonstrating simulation composition

The presence of noise is currently one of the main obstacles to achieving large-scale quantum computation. Strategies to characterise and understand noise processes in quantum hardware are a critical part of mitigating it, especially as the overhead of full error correction and fault-tolerance is beyond the reach of current hardware. Non-Markovian effects are a particularly unfavourable type of noise, being both harder to analyse using standard techniques and more difficult to control using error correction. In this work we develop a set of efficient algorithms, based on the rigorous mathematical theory of Markovian master equations, to analyse and evaluate unknown noise processes. In the case of dynamics consistent with Markovian evolution, our algorithm outputs the best-fit Lindbladian, i.e., the generator of a memoryless quantum channel which best approximates the tomographic data to within the given precision. In the case of non-Markovian dynamics, our algorithm returns a quantitative and operationally meaningful measure of non-Markovianity in terms of isotropic noise addition. We provide a Python implementation of all our algorithms, and benchmark these on a range of 1- and 2-qubit examples of synthesised noisy tomography data, generated using the Cirq platform. The numerical results show that our algorithms succeed both in extracting a full description of the best-fit Lindbladian to the measured dynamics, and in computing accurate values of non-Markovianity that match analytical calculations., Comment: published version, 36 + 24 pages, merging material from arXiv:2303.08936. Code available at: https://gitlab.com/TamaraKohler/non-markovianity

Recent work has demonstrated the existence of universal Hamiltonians - simple spin lattice models that can simulate any other quantum many body system to any desired level of accuracy. Until now proofs of universality have relied on explicit constructions, tailored to each specific family of universal Hamiltonians. In this work we go beyond this approach, and completely classify the simulation ability of quantum Hamiltonians by their complexity classes. We do this by deriving necessary and sufficient complexity theoretic conditions characterising universal quantum Hamiltonians. Although the result concerns the theory of analogue Hamiltonian simulation - a promising application of near-term quantum technology - the proof relies on abstract complexity theoretic concepts and the theory of quantum computation. As well as providing simplified proofs of previous Hamiltonian universality results, and offering a route to new universal constructions, the results in this paper give insight into the origins of universality. For example, finally explaining the previously noted coincidences between families of universal Hamiltonian and classes of Hamiltonians appearing in complexity theory., Comment: 22 pages. V2 - extended discussion, fixing typos

Recent work has characterised rigorously what it means for one quantum system to simulate another, and demonstrated the existence of universal Hamiltonians -- simple spin lattice Hamiltonians that can replicate the entire physics of any other quantum many body system. Previous universality results have required proofs involving complicated `chains' of perturbative `gadgets'. In this paper, we derive a significantly simpler and more powerful method of proving universality of Hamiltonians, directly leveraging the ability to encode quantum computation into ground states. This provides new insight into the origins of universal models, and suggests a deep connection between universality and complexity. We apply this new approach to show that there are universal models even in translationally invariant spin chains in 1D. This gives as a corollary a new Hamiltonian complexity result, that the local Hamiltonian problem for translationally-invariant spin chains in one dimension with an exponentially-small promise gap is PSPACE-complete. Finally, we use these new universal models to construct the first known toy model of 2D--1D holographic duality between local Hamiltonians., Comment: 31 pages. V2: extended overview of proof written to make paper clearer. This version of the article has been accepted for publication after peer review, but is not the version of record and does not reflect any post acceptance improvements. The version of record is available online at http://dx.doi.org/10.1007/s00023-021-01111-7

Holographic quantum error correcting codes (HQECC) have been proposed as toy models for the AdS/CFT correspondence, and exhibit many of the features of the duality. HQECC give a mapping of states and observables. However, they do not map local bulk Hamiltonians to local Hamiltonians on the boundary. In this work, we combine HQECC with Hamiltonian simulation theory to construct a bulk-boundary mapping between local Hamiltonians, whilst retaining all the features of the HQECC duality. This allows us to construct a duality between models, encompassing the relationship between bulk and boundary energy scales and time dynamics. It also allows us to construct a map in the reverse direction: from local boundary Hamiltonians to the corresponding local Hamiltonian in the bulk. Under this boundary-to-bulk mapping, the bulk geometry emerges as an approximate, low-energy, effective theory living in the code-space of an (approximate) HQECC on the boundary. At higher energy scales, this emergent bulk geometry is modified in a way that matches the toy models of black holes proposed previously for HQECC. Moreover, the duality on the level of dynamics shows how these toy-model black holes can form dynamically., Comment: 46 + 14 pages. v2: extended discussion. v3: extended introduction to make proof overview & main results more accessible, minor typos fixed

Spin models are widely studied in the natural sciences, from investigating magnetic materials in condensed matter physics to studying neural networks. Previous work has demonstrated that there exist simple classical spin models that are universal: they can replicate -- in a precise and rigorous sense -- the complete physics of any other classical spin model, to any desired accuracy. However, all previously known universal models break translational invariance. In this paper we show that there exist translationally invariant universal models. Our main result is an explicit construction of a translationally invariant, 2D, nearest-neighbour, universal classical Hamiltonian with a single free parameter. The proof draws on techniques from theoretical computer science, in particular recent complexity theoretic results on tiling problems. Our results imply that there exists a single Hamiltonian which encompasses all classical spin physics, just by tuning a single parameter and varying the size of the lattice. We also prove that our construction is optimal in terms of the number of parameters in the Hamiltonian; there cannot exist a translationally invariant universal Hamiltonian with only the lattice size as a parameter., Comment: 38 pages, 3 figures. v2: Extended discussion, simplified overview