We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle). Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum' particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the analogous classical problem, one dimensional MAX-2-SAT with nearest neighbor constraints, which is in P.The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Some illegal configurations cannot be ruled out by local checks, and are instead ruled out because they would, in the future, evolve into a state which can be seen locally to be illegal. Assuming BQP ne QMA, our construction gives a one-dimensional system which takes an exponential time to relax to its ground state at any temperature. This makes it a candidate for a one-dimensional spin glass.We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a one-dimensional quantum system (with 9 states per particle). Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum' particles is QMA-complete; QMA is a quantum analogue of NP. This is in striking contrast to the analogous classical problem, one dimensional MAX-2-SAT with nearest neighbor constraints, which is in P.The proof of the QMA-completeness result requires an additional idea beyond the usual techniques in the area: Some illegal configurations cannot be ruled out by local checks, and are instead ruled out because they would, in the future, evolve into a state which can be seen locally to be illegal. Assuming BQP ne QMA, our construction gives a one-dimensional system which takes an exponential time to relax to its ground state at any temperature. This makes it a candidate for a one-dimensional spin glass.

Annual Symposium on Foundations of Computer ScienceOct 1, 200740 citations

The Complexity of Quantum Systems on a One-dimensional Chain

Sandy Irani

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 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.

arXiv:0705.4067May 28, 200712 citations

The computational power of many-body systems

Zachary Webb

Many-body systems are well known throughout physics to be hard problems to exactly solve, but much of this is folklore resulting from the lack of an analytic solution to these systems. This thesis attempts to classify the complexity inherent in many of these systems, and give quantitative results for why the problems are hard. In particular, we analyze the many-particle system corresponding to a multi-particle quantum walk, showing that the time evolution of such systems on a polynomial sized graph is universal for quantum computation, and thus determining how a particular state evolves is as hard as an arbitrary quantum computation. We then analyze the ground energy properties of related systems, showing that for bosons, bounding the ground energy of the same Hamiltonian with a fixed number of particles is QMA-complete. Similar techniques provide a novel proof that quantum walk is universal for quantum computing, and constructs a QMA-complete problem that does not reference quantum mechanics.Many-body systems are well known throughout physics to be hard problems to exactly solve, but much of this is folklore resulting from the lack of an analytic solution to these systems. This thesis attempts to classify the complexity inherent in many of these systems, and give quantitative results for why the problems are hard. In particular, we analyze the many-particle system corresponding to a multi-particle quantum walk, showing that the time evolution of such systems on a polynomial sized graph is universal for quantum computation, and thus determining how a particular state evolves is as hard as an arbitrary quantum computation. We then analyze the ground energy properties of related systems, showing that for bosons, bounding the ground energy of the same Hamiltonian with a fixed number of particles is QMA-complete. Similar techniques provide a novel proof that quantum walk is universal for quantum computing, and constructs a QMA-complete problem that does not reference quantum mechanics.

Sep 28, 20160 citations

Universal Qudit Hamiltonians

Stephen Piddock, Ashley Montanaro

Abstract A family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits (d-level systems): We completely characterise the k-qudit interactions which are universal, if augmented with arbitrary Hermitian 1-local terms. We find that, for all $k \geqslant 2$ k ⩾ 2 and all local dimensions $d \geqslant 2$ d ⩾ 2 , almost all such interactions are universal aside from a simple stoquastic class. We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the SU(d) and SU(2) Heisenberg interactions are universal for all local dimensions $d \geqslant 2$ d ⩾ 2 (spin $\geqslant 1/2$ ⩾ 1 / 2 ), implying that a quantum variant of the Max-d-Cut problem is QMA-complete. We also show that for $d=3$ d = 3 all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model. We prove universality of any interaction proportional to the projector onto a pure entangled state.Abstract A family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits (d-level systems): We completely characterise the k-qudit interactions which are universal, if augmented with arbitrary Hermitian 1-local terms. We find that, for all $$k \geqslant 2$$ k ⩾ 2 and all local dimensions $$d \geqslant 2$$ d ⩾ 2 , almost all such interactions are universal aside from a simple stoquastic class. We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the SU(d) and SU(2) Heisenberg interactions are universal for all local dimensions $$d \geqslant 2$$ d ⩾ 2 (spin $$\geqslant 1/2$$ ⩾ 1 / 2 ), implying that a quantum variant of the Max-d-Cut problem is QMA-complete. We also show that for $$d=3$$ d = 3 all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model. We prove universality of any interaction proportional to the projector onto a pure entangled state.

arXiv:1802.07130Communications in Mathematical PhysicsFeb 23, 20217 citations

Adiabatic Quantum Computation

Friederike Anna Dziemba

The quantum adiabatic theorem ensures that a slowly changing system, initially prepared in its ground state, will evolve to its final ground state with arbitrary precision. As a first result this thesis extends the original theorem to projection operators keeping the statement valid for Hamiltonians with degenerate ground spaces. Yet the main focus of this work lies in studying the efficiency of quantum circuit simulations by adabatic quantum computation. The standard Hamiltonian construction by Kitaev is based on a path graph reflecting the $L$ computation steps and influencing the scaling of the necessary evolution time by its spectral gap of $\mathcal{O}\left(\frac{1}{L^2}\right)$ . Aspiring to an improved running time we generalize Kitaev's Hamiltonian to so-called standard graph Hamiltonians based on graph families with a different spectral gap. In this generalized construction the first two time derivatives of the Hamiltonian and the fraction of initial vertices appear as additional parameters of running time. In a first step the time derivatives can be proven to be constant. Expansion results from spectral graph theory however impose a trade-off between the spectral gap and the fraction of initial vertices as well as the fraction of final vertices which corresponds to the probability for obtaining the correct computational output. Graphs with spectral gap $\mathcal{O}\left(\frac{1}{L^k}\right)$ , $k<2$ , turn out to contradict very likely, graphs with $k<1$ even for sure at least one of the efficiency criterias for running time, output probability or Hamiltonian implementation. The above results may also be obstacles for a possible quantum PCP-theorem in complexity theory claiming the local Hamiltonian problem with constant gap to be QMA-complete since the very same Kitaev Hamiltonian is constructed in the QMA-hardness proof for the Local Hamiltonian problem.The quantum adiabatic theorem ensures that a slowly changing system, initially prepared in its ground state, will evolve to its final ground state with arbitrary precision. As a first result this thesis extends the original theorem to projection operators keeping the statement valid for Hamiltonians with degenerate ground spaces. Yet the main focus of this work lies in studying the efficiency of quantum circuit simulations by adabatic quantum computation. The standard Hamiltonian construction by Kitaev is based on a path graph reflecting the $L$ computation steps and influencing the scaling of the necessary evolution time by its spectral gap of $\mathcal{O}\left(\frac{1}{L^2}\right)$. Aspiring to an improved running time we generalize Kitaev's Hamiltonian to so-called standard graph Hamiltonians based on graph families with a different spectral gap. In this generalized construction the first two time derivatives of the Hamiltonian and the fraction of initial vertices appear as additional parameters of running time. In a first step the time derivatives can be proven to be constant. Expansion results from spectral graph theory however impose a trade-off between the spectral gap and the fraction of initial vertices as well as the fraction of final vertices which corresponds to the probability for obtaining the correct computational output. Graphs with spectral gap $\mathcal{O}\left(\frac{1}{L^k}\right)$, $k<2$, turn out to contradict very likely, graphs with $k<1$ even for sure at least one of the efficiency criterias for running time, output probability or Hamiltonian implementation. The above results may also be obstacles for a possible quantum PCP-theorem in complexity theory claiming the local Hamiltonian problem with constant gap to be QMA-complete since the very same Kitaev Hamiltonian is constructed in the QMA-hardness proof for the Local Hamiltonian problem.

arXiv:1610.04708Oct 15, 20161 citation

The complexity of the consistency and N-representability problems for quantum states - eScholarship

Yi-Kai Liu

Quantum mechanics has important consequences for machines that store and manipulate information. In particular, quantum computers might be more powerful than classical computers; examples of this include Shor's algorithm for factoring and discrete logarithms, and Grover's algorithm for black-box search. Because of these theoretical results, and the possibility that we may eventually succeed in building scalable quantum computers, it is interesting to study complexity classes based on quantum computation. QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most of which are variants of the Local Hamiltonian problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems which are very different from those known previously, and have applications in quantum chemistry. The first one is Consistency of Local Density Matrices : given a collection of density matrices describing different subsets of an n-qubit system (where each subset has constant size), decide whether these are consistent with some global state of all n qubits. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. Our reduction is based on algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, Fermionic Local Hamiltonian and N- representability, are QMA-complete. These problems involve systems of fermions, rather than qubits; they arise in calculating the ground state energies of molecular systems. N-representability is particularly interesting, as it is a key component in recently developed numerical methods using the contracted Schrodinger equation. Although these problems have been studied since the 1960's, it is only recently that the theory of quantum computation has provided the right tools to properly characterize their complexity. Finally, we study some special cases of the Consistency problem, pertaining to 1-dimensional and stoquastic systems. We also give an alternative proof of a result due to Jaynes : whenever local density matrices are consistent, they are consistent with a Gibbs stateQuantum mechanics has important consequences for machines that store and manipulate information. In particular, quantum computers might be more powerful than classical computers; examples of this include Shor's algorithm for factoring and discrete logarithms, and Grover's algorithm for black-box search. Because of these theoretical results, and the possibility that we may eventually succeed in building scalable quantum computers, it is interesting to study complexity classes based on quantum computation. QMA (Quantum Merlin-Arthur) is the quantum analogue of the class NP. There are a few QMA-complete problems, most of which are variants of the Local Hamiltonian problem introduced by Kitaev. In this dissertation we show some new QMA-complete problems which are very different from those known previously, and have applications in quantum chemistry. The first one is Consistency of Local Density Matrices : given a collection of density matrices describing different subsets of an n-qubit system (where each subset has constant size), decide whether these are consistent with some global state of all n qubits. This problem was first suggested by Aharonov. We show that it is QMA-complete, via an oracle reduction from Local Hamiltonian. Our reduction is based on algorithms for convex optimization with a membership oracle, due to Yudin and Nemirovskii. Next we show that two problems from quantum chemistry, Fermionic Local Hamiltonian and N- representability, are QMA-complete. These problems involve systems of fermions, rather than qubits; they arise in calculating the ground state energies of molecular systems. N-representability is particularly interesting, as it is a key component in recently developed numerical methods using the contracted Schrodinger equation. Although these problems have been studied since the 1960's, it is only recently that the theory of quantum computation has provided the right tools to properly characterize their complexity. Finally, we study some special cases of the Consistency problem, pertaining to 1-dimensional and stoquastic systems. We also give an alternative proof of a result due to Jaynes : whenever local density matrices are consistent, they are consistent with a Gibbs state

Jan 1, 20070 citations

k \geqslant 2

k ⩾ 2 and all local dimensions

d \geqslant 2

d ⩾ 2 , almost all such interactions are universal aside from a simple stoquastic class. We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the SU(d) and SU(2) Heisenberg interactions are universal for all local dimensions

d \geqslant 2

d ⩾ 2 (spin

\geqslant 1/2

⩾ 1 / 2 ), implying that a quantum variant of the Max-d-Cut problem is QMA-complete. We also show that for

d=3

d = 3 all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model. We prove universality of any interaction proportional to the projector onto a pure entangled state.

L

computation steps and influencing the scaling of the necessary evolution time by its spectral gap of

\mathcal{O}\left(\frac{1}{L^2}\right)

. Aspiring to an improved running time we generalize Kitaev's Hamiltonian to so-called standard graph Hamiltonians based on graph families with a different spectral gap. In this generalized construction the first two time derivatives of the Hamiltonian and the fraction of initial vertices appear as additional parameters of running time. In a first step the time derivatives can be proven to be constant. Expansion results from spectral graph theory however impose a trade-off between the spectral gap and the fraction of initial vertices as well as the fraction of final vertices which corresponds to the probability for obtaining the correct computational output. Graphs with spectral gap

\mathcal{O}\left(\frac{1}{L^k}\right)

k<2

, turn out to contradict very likely, graphs with

k<1

even for sure at least one of the efficiency criterias for running time, output probability or Hamiltonian implementation. The above results may also be obstacles for a possible quantum PCP-theorem in complexity theory claiming the local Hamiltonian problem with constant gap to be QMA-complete since the very same Kitaev Hamiltonian is constructed in the QMA-hardness proof for the Local Hamiltonian problem.