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arXiv:2210.06796

Circuit depth versus energy in topologically ordered systems

Arkin Tikku, Isaac H. Kim

Published Oct 13, 2022• 0 citations
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Abstract

We prove a nontrivial circuit-depth lower bound for preparing a low-energy state of a locally interacting quantum many-body system in two dimensions, assuming the circuit is geometrically local. For preparing any state which has an energy density of at most $ε$ with respect to Kitaev's toric code Hamiltonian on a two dimensional lattice $Λ$, we prove a lower bound of $Ω\left(\min\left(1/ε^{\frac{1-α}{2}}, \sqrt{|Λ|}\right)\right)$ for any $α>0$. We discuss two implications. First, our bound implies that the lowest energy density obtainable from a large class of existing variational circuits (e.g., Hamiltonian variational ansatz) cannot, in general, decay exponentially with the circuit depth. Second, if long-range entanglement is present in the ground state, this can lead to a nontrivial circuit-depth lower bound even at nonzero energy density. Unlike previous approaches to prove circuit-depth lower bounds for preparing low energy states, our proof technique does not rely on the ground state to be degenerate.

DOI
10.48550/arxiv.2210.06796

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We prove a nontrivial circuit-depth lower bound for preparing a low-energy state of a locally interacting quantum many-body system in two dimensions, assuming the circuit is geometrically local. For preparing any state which has an energy density of at most εεε with respect to Kitaev's toric code Hamiltonian on a two dimensional lattice ΛΛΛ, we prove a lower bound of Ω(min⁡(1/ε1−α2,∣Λ∣))Ω\left(\min\left(1/ε^{\frac{1-α}{2}}, \sqrt{|Λ|}\right)\right)Ω(min(1/ε21−α​,∣Λ∣​)) for any α>0α>0α>0. We discuss two implications. First, our bound implies that the lowest energy density obtainable from a large class of existing variational circuits (e.g., Hamiltonian variational ansatz) cannot, in general, decay exponentially with the circuit depth. Second, if long-range entanglement is present in the ground state, this can lead to a nontrivial circuit-depth lower bound even at nonzero energy density. Unlike previous approaches to prove circuit-depth lower bounds for preparing low energy states, our proof technique does not rely on the ground state to be degenerate.
This thesis concerns itself with two fundamental tasks in the theory of quantum information: the preparation of ground and low-energy states of local Hamiltonians on a quantum computer and the secure transmission of classical information over a quantum channel. The first part of this thesis addresses the former task and is dedicated in its entirety to quantum many-body physics. We will be focusing on topologically ordered lattice models in two spatial dimensions. It is known that the presence of long-range entanglement in the ground states of these models necessitates local quantum circuits of depth linear in the diameter of the system for their preparation. We study the consequences of this hardness result for the preparation of low-energy states of these models. In particular we show that preparing such states requires local quantum circuits of polynomial depth in the inverse energy density of the states and comment on the relevance of these results to quantum computation in the near term. Our technical contribution is to give a circuit depth lower bound for low-energy states independent of the dimension of the ground space. We then experimentally demonstrate how currently existing ion trap hardware has reached a level of maturity where noisy adaptive circuits of constant depth outperform noiseless, non-adaptive circuits of the same depth for the task of approximating a ground state of the the toric code. In the last part of this thesis, we switch gears to quantum Shannon theory and study private communication over a classical-quantum wiretap channel. We study the problem of non-additivity of the private information for this model. Surprisingly, we find that the private information is non-additive when either of the outputs of the channel is allowed to be quantum, while the input is classical. However, there exists a single-letter formula when the input state is quantum and the outputs are classical due to the private information of the channel becoming additive.
The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings, 2014 -- which posits the existence of a local Hamiltonian with a super-constant quantum circuit lower bound on the complexity of all low-energy states -- identifies a fundamental obstacle to the resolution of the quantum PCP conjecture. In this work, we provide new techniques, based on entropic and local indistinguishability arguments, that prove circuit lower bounds for all the low-energy states of local Hamiltonians arising from quantum error-correcting codes. For local Hamiltonians arising from nearly linear-rate or nearly linear-distance LDPC stabilizer codes, we prove super-constant circuit lower bounds for the complexity of all states of energy o(n). Such codes are known to exist and are not necessarily locally testable, a property previously suspected to be essential for the NLTS conjecture. Curiously, such codes can also be constructed on a two-dimensional lattice, showing that low-depth states cannot accurately approximate the ground-energy even in physically relevant systems.
A central question in quantum information theory is the circuit complexity of states arising from standard many-body models. We study this question for quantum ppp-spin glasses, random Hamiltonians whose interactions act on ppp-tuples of qubits through Pauli strings. Anschuetz, Gamarnik, and Kiani (arXiv:2404.07231) showed that the optimum energy is separated from the best energy achievable by product states. This leaves open whether shallow circuits can close the gap, since even depth-one circuits can generate entanglement. We show that the entanglement needed to close the product-state gap cannot be generated at shallow depth. When the average interaction degree grows with nnn, we prove that, for all sufficiently large fixed ppp, any circuit preparing an nnn-qubit state whose normalized energy is within a fixed positive constant of the optimum must have depth Ωp(log⁡n)Ω_p(\log n)Ωp​(logn). In the bounded-average-degree regime, we prove a fixed-depth obstruction: for every fixed DDD, a sufficiently large degree prefactor rules out depth-DDD preparation of near-ground states. Both results hold uniformly over circuits with an arbitrary number of ancilla qubits. Our results give an obstruction in the spirit of the No Low-Energy Trivial States problem of Freedman and Hastings (arXiv:1301.1363), but for random quantum spin glasses rather than code-based Hamiltonians such as those of Anshu, Breuckmann, and Nirkhe (arXiv:2206.13228), whose ground states admit polynomial-size preparation circuits. This setting opens a probabilistic route to NLTS-like questions: we recast state-preparation lower bounds for random quantum Hamiltonians as uniform control of Gaussian processes indexed by shallow circuits.
We show that the low-energy states of non-Abelian topological orders possess extensive magic which is long-ranged, and cannot be eliminated by a constant-depth local unitary circuit. This refines conventional notions of complexity beyond the linear circuit depth which is required to prepare any topological phase, and provides a new resource-theoretic characterization of topological orders. A central technical result is a no-go theorem establishing that stabilizer states--even up to constant-depth local unitarie--cannot approximate low-energy states of non-Abelian string-net models which satisfy the entanglement bootstrap axioms. Moreover, we show that stabilizer-realizable Abelian string-net phases have mutual braiding phases quantized by the on-site qudit dimension, and that any violation of this condition necessarily implies extensive long-range magic. Extending to higher spatial dimensions, we argue that any state obeying an entanglement area law and hosting excitations with nontrivial fusion spaces must exhibit extensive long-range magic. This applies, in particular, to ground-states and low-energy states of higher-dimensional quantum double models.
We provide an Ω(log(n))Ω(log(n))Ω(log(n)) lower bound for the depth of any quantum circuit generating the unique groundstate of Kitaev's spherical code. No circuit-depth lower bound was known before on this code in the general case where the gates can connect qubits even if they are far away; To the best of our knowledge, this is the first time a quantum circuit-depth lower bound is given for unique ground state of a {\it gapped} local Hamiltonian. Providing a lower bound in this case seems more challenging, since such systems exhibit exponential decay of correlations and standard lower bound techniques do not apply. We prove our lower bound by introducing the new notion of γγγ-separation, and analyzing its behavior using algebraic topology arguments. We extend out methods also to a wide class of polygonal complexes beyond the sphere, and prove a circuit-depth lower bound whenever the complex does not have a small "bottle neck" (in a sense which we define). Here our lower bound on the circuit depth is only Ω(logloglog(n))Ω(logloglog(n))Ω(logloglog(n)). We conjecture that the correct lower bound is at least Ω(log(log(n))Ω(log(log(n))Ω(log(log(n)), but this seems harder to achieve due to the possibility of hyperbolic geometry. For general simplicial complexes the lack of geometrical restriction on the gates becomes considerably more problematic than for the sphere, and we need to thoroughly modify the original argument in order to get a meaningful bound. To the best of our knowledge, this is the first time the class of trivial quantum states is separated from the class of unique ground states of gapped local Hamiltonians; we provide a survey of the current status of this hierarchy for completeness.The tools developed here will be useful in various contexts in which quantum circuit depth lower bounds are of interest, including the study of topological order, quantum computational complexity and quantum algorithmic speedups.
Ground state preparation is classically intractable for general Hamiltonians. On quantum devices, shallow parametrized circuits can be effectively trained to obtain short-range entangled states under the paradigm of variational quantum eigensolver, while deep circuits are generally untrainable due to the barren plateau phenomenon. In this Letter, we give a general lower bound on the variance of circuit gradients for arbitrary quantum circuits composed of local 2-designs. Based on our unified framework, we prove the absence of barren plateaus in training finite local-depth circuits (FLDC) for the ground states of local Hamiltonians. FLDCs are allowed to be deep in the conventional circuit depth to generate long-range entangled ground states, such as topologically ordered states, but their local depths are finite, i.e., there is only a finite number of gates acting on individual qubits. This characteristic sets FLDC apart from shallow circuits: FLDC in general cannot be classically simulated to estimate local observables efficiently by existing tensor network methods in two and higher dimensions. We validate our analytical results with extensive numerical simulations and demonstrate the effectiveness of variational training using the generalized toric code model.