The Complexity of Quantum Systems on a One-dimensional Chain
Sandy Irani
备注：
This paper has been merged with arXiv:0705.4077 and is now co-authored with Dorit Aharonov, Daniel Gottesman, and Julia Kempe. Ther version posted here is the same as the original version
We prove that adiabatic computation is equivalent to standard quantum computation even when the adiabatic quantum system is restricted to be a set of particles on a one-dimensional chain. We give a construction that uses a 2-local Hamiltonian on nearest neighbors using particles that can have ten distinct states. This implies a construction of a one-dimensional chain of qubits in which the Hamiltonian is 6-local. We adapt this construction to show that the 2-local Hamiltonian for 13-state particles is QMA-complete which in turn implies that the 8-local Hamiltonian restricted to a one-dimensional chain of qubits is QMA-complete.
We develop a theory of classical complexity. We study the relations between classical complexity and entropy, and conjecture that in an isolated system, classical absolute complexity always tends to grow, until it reaches its maximum. We calculate some exact closed-form expressions of the growth of average classical complexity over time in some concrete models, and gain further insights of both classical and quantum complexity by using the theory of Markov chains.
Minimum-weight perfect matching (MWPM) has been been the primary classical algorithm for error correction in the surface code, since it is of low runtime complexity and achieves relatively low logical error rates [Phys. Rev. Lett. 108, 180501 (2012)]. A Markov chain Monte Carlo (MCMC) algorithm [Phys. Rev. Lett. 109, 160503 (2012)] is able to achieve lower logical error rates and higher thresholds than MWPM, but requires a classical runtime complexity which is super-polynomial in L, the linear size of the code. In this work we present an MCMC algorithm that achieves significantly lower logical…
The minimal memory required to model a given stochastic process - known as the statistical complexity - is a widely adopted quantifier of structure in complexity science. Here, we ask if quantum mechanics can fundamentally change the qualitative behaviour of this measure. We study this question in the context of the classical Ising spin chain. In this system, the statistical complexity is known to grow monotonically with temperature. We evaluate the spin chain's quantum mechanical statistical complexity by explicitly constructing its provably simplest quantum model, and demonstrate that this m…
We study non-equilibrium steady states (NESS) of spin chains with boundary Markovian dissipation from the computational complexity point of view. We focus on XX chains whose NESS are matrix product operators (MPO), i.e. with coefficients of a tensor operator basis described by transition amplitudes in an auxiliary space. Encoding quantum algorithms in the auxiliary space, we show that estimating expectations of operators, being local in the sense that each acts on disjoint sets of few spins covering all the system, provides the answers of problems at least as hard as, and believed by many comp…
A number of recent works have argued that quantum complexity, a well-known concept in computer science that has re-emerged recently in the context of the physics of black holes, may be used as an efficient probe of novel phenomena such as quantum chaos and even quantum phase transitions. In this article, we provide further support for the latter, using a 1-dimensional p-wave superconductor - the Kitaev chain - as a prototype of a system displaying a topological phase transition. The Hamiltonian of the Kitaev chain manifests two gapped phases of matter with fermion parity symmetry; a trivial st…
The simulation complexity of predicting the time evolution of delocalized many-body quantum systems has attracted much recent interest, and simulations of such systems in real quantum hardware are promising routes to demonstrating a quantum advantage over classical machines. In these proposals, random noise is an obstacle that must be overcome for a faithful simulation, and a single error event can be enough to drive the system to a classically trivial state. We argue that this need not always be the case, and consider a modification to a leading quantum sampling problem-- time evolution in an…
Quantum speedup for mixing a Markov chain can be achieved based on the construction of slowly-varying $r$ Markov chains where the initial chain can be easily prepared and the spectral gaps have uniform lower bound. The overall complexity is proportional to $r$. We present a multi-level approach to construct such a sequence of $r$ Markov chains by varying a resolution parameter $h.$ We show that the density function of a low-resolution Markov chain can be used to warm start the Markov chain with high resolution. We prove that in terms of the chain length the new algorithm has $O(1)$ complexity …
This work recollects a non-universal set of quantum gates described by higher-dimensional Spin groups. They are also directly related with matchgates in theory of quantum computations and complexity. Various processes of quantum state distribution along a chain such as perfect state transfer and different types of quantum walks can be effectively modeled on classical computer using such approach.
Genuine multipartite entanglement has been found in some spin chain systems. However, genuine multipartite nonlocality, which is much rarer than genuine multipartite entanglement, has never been found in any spin chain system. Here we present genuine multipartite nonlocality in a spin chain system. After introducing the definition of genuine multipartite nonlocality and a multipartite Bell-type inequality, we construct a group of joint measurements for the inequality in a one-dimensional ferromagnetic $N$-qubit chain with nearest-neighbor XXZ interaction, and many violations to the inequality …
The leakage chain rule for quantum min-entropy quantifies the change of min-entropy when one party gets additional leakage about the information source. Herein we provide an interactive version that quantifies the change of min-entropy between two parties, who share an initial classical-quantum state and are allowed to run a two-party protocol. As an application, we prove new versions of lower bounds on the complexity of quantum communication of classical information.
In this paper, we investigate a high-dimensional quantum state transfer protocol. An arbitrary unknown high-dimensional state can be transferred with high fidelity between two remote registers through a XX coupling spin chain of arbitrary length. The evolution of the state transfer is determined by the natural dynamics of the chain without external modulation and coupling strength engineering. As a consequence, entanglement distribution with high efficiency can be achieved. Also the strong field and high spin quantum number can counteract partly the effect of finite temperature to ensure high …
We extend the notion of Gacs quantum algorithmic entropy, originally formulated for finitely many qubits, to infinite dimensional quantum spin chains and investigate the relation of this extension with two quantum dynamical entropies that have been proposed in recent years. Further, we prove an extension of Brudno's theorem in quantum spin chains with shift dynamics.
We study the temporal evolution of the circuit complexity after the local quench where two harmonic chains are suddenly joined, choosing the initial state as the reference state. We discuss numerical results for the complexity for the entire chain and the subsystem complexity for a block of consecutive sites, obtained by exploiting the Fisher information geometry of the covariance matrices. The qualitative behaviour of the temporal evolutions of the subsystem complexity depends on whether the joining point is inside the subsystem. The revivals and a logarithmic growth observed during these tem…
We study the temporal evolution of the circuit complexity for a subsystem in harmonic lattices after a global quantum quench of the mass parameter, choosing the initial reduced density matrix as the reference state. Upper and lower bounds are derived for the temporal evolution of the complexity for the entire system. The subsystem complexity is evaluated by employing the Fisher information geometry for the covariance matrices. We discuss numerical results for the temporal evolutions of the subsystem complexity for a block of consecutive sites in harmonic chains with either periodic or Dirichle…
We prove that estimating the ground state energy of a translationally-invariant, nearest-neighbour Hamiltonian on a 1D spin chain is QMAEXP-complete, even for systems of low local dimension (roughly 40). This is an improvement over the best previously-known result by several orders of magnitude, and it shows that spin-glass-like frustration can occur in translationally-invariant quantum systems with a local dimension comparable to the smallest-known non-translationally-invariant systems with similar behaviour. While previous constructions of such systems rely on standard models of quantum comp…
A complete classification is given for one dimensional chains with nearest neighbor interactions having two states in each site, for which a matrix product ground state exists. The Hamiltonians and their corresponding matrix product ground states are explicitly obtained.
Excited states of spin-chains play an important role in condensed matter physics. We present a method of calculating the single magnon excited states of the Heisenberg spin-chain that can be efficiently implemented on a quantum processor for small spin chains. Our method involves finding the stationary points of the energy vs wavenumber curve. We implement our method for 4-site and 8-site Heisenberg Hamiltonians using numerical techniques as well as using an IBM quantum processor. Finally, we give an insight into the circuit complexity and scaling of our proposed method.
We use Nielsen's geometric approach to quantify the circuit complexity in a one-dimensional Kitaev chain across a topological phase transition. We find that the circuit complexities of both the ground states and non-equilibrium steady states of the Kitaev model exhibit non-analytical behaviors at the critical points, and thus can be used to detect both {\it equilibrium} and {\it dynamical} topological phase transitions. Moreover, we show that the locality property of the real-space optimal Hamiltonian connecting two different ground states depends crucially on whether the two states belong to …
Quantum complexity, suitably defined, has been suggested as an important probe of late-time dynamics of black holes, particularly in the context of AdS/CFT. A notion of quantum complexity can be effectively captured by quantifying the spread of an operator in Krylov space as a consequence of time evolution. Complexity is expected to behave differently in chaotic many-body systems, as compared to integrable ones. In this paper we investigate Krylov complexity for the case of interacting integrable models at finite size and find that complexity saturation is suppressed as compared to chaotic sys…
