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M. B. Hastings

Publications and source records attributed to M. B. Hastings.

At least 37 records · Page 2Linked to original sources
arXiv2017

Distillation with sublogarithmic overhead

It has been conjectured [1] that for any distillation protocol for magic states for the $T$ gate, the number of noisy input magic states required per output magic state at output error rate $ε$ is $Ω(\log(1/ε))$. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters $[[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]$ for any integers $ m > 2r$, $r > w \ge 0$, by puncturing quantum Reed-Muller codes. When $m > νr$, our code admits a transversal logical gate at the $ν$-th level of Clifford hierarchy. In a distillation protocol for magic states at the level $ν= 3$ ($T$-gate), the ratio of input to output magic states is $O(\log^γ(1/ε))$ where $γ= \log(n/k)/\log(d)< 0.678$ for some $m,r,w$. The smallest code in our family for which $γ< 1$ is on $\approx 2^{58}$ qubits.

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arXiv2017

The Stability of Free Fermi Hamiltonians

Recent results have shown the stability of frustration-free Hamiltonians to weak local perturbations, assuming several conditions. In this paper, we prove the stability of free fermion Hamiltonians which are gapped and local. These free fermion Hamiltonians are not necessarily frustration-free, but we are able to adapt previous work to prove stability. The key idea is to add an additional copy of the system to cancel topological obstructions. We comment on applications to quantization of Hall conductance in such systems.

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arXiv2017

Small Majorana Fermion Codes

We consider Majorana fermion stabilizer codes with small number of modes and distance. We give an upper bound on the number of logical qubits for distance $4$ codes, and we construct Majorana fermion codes similar to the classical Hamming code that saturate this bound. We perform numerical studies and find other distance $4$ and $6$ codes that we conjecture have the largest possible number of logical qubits for the given number of physical Majorana modes. Some of these codes have more logical qubits than any Majorana fermion code derived from a qubit stabilizer code.

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arXiv2016

Turning Gate Synthesis Errors into Incoherent Errors

Using error correcting codes and fault tolerant techniques, it is possible, at least in theory, to produce logical qubits with significantly lower error rates than the underlying physical qubits. Suppose, however, that the gates that act on these logical qubits are only approximation of the desired gate. This can arise, for example, in synthesizing a single qubit unitary from a set of Clifford and $T$ gates; for a generic such unitary, any finite sequence of gates only approximates the desired target. In this case, errors in the gate can add coherently so that, roughly, the error $ε$ in the unitary of each gate must scale as $ε\lesssim 1/N$, where $N$ is the number of gates. If, however, one has the option of synthesizing one of several unitaries near the desired target, and if an average of these options is closer to the target, we give some elementary bounds showing cases in which the errors can be made to add incoherently by averaging over random choices, so that, roughly, one needs $ε\lesssim 1/\sqrt{N}$. We remark on one particular application to distilling magic states where this effect happens automatically in the usual circuits.

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arXiv2016

Weight Reduction for Quantum Codes

We present an algorithm that takes a CSS stabilizer code as input, and outputs another CSS stabilizer code such that the stabilizer generators all have weights $O(1)$ and such that $O(1)$ generators act on any given qubit. The number of logical qubits is unchanged by the procedure, while we give bounds on the increase in number of physical qubits and in the effect on distance and other code parameters, such as soundness (as a locally testable code) and "cosoundness" (defined later). Applications are discussed, including to codes from high-dimensional manifolds which have logarithmic weight stabilizers. Assuming a conjecture in geometry\cite{hdm}, this allows the construction of CSS stabilizer codes with generator weight $O(1)$ and almost linear distance. Another application of the construction is to increasing the distance to $X$ or $Z$ errors, whichever is smaller, so that the two distances are equal.

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arXiv2016

Local Maxima and Improved Exact Algorithm for MAX-2-SAT

Given a MAX-2-SAT instance, we define a local maximum to be an assignment such that changing any single variable reduces the number of satisfied clauses. We consider the question of the number of local maxima that an instance of MAX-2-SAT can have. We give upper bounds in both the sparse and nonsparse case, where the sparse case means that there is a bound $d$ on the average number of clauses involving any given variable. The bounds in the nonsparse case are tight up to polylogarithmic factors, while in the sparse case the bounds are tight up to a multiplicative factor in $d$ for large $d$. Additionally, we generalize to the question of assignments which are maxima up to changing $k> 1$ variables simultaneously; in this case, we give explicit constructions with large (in a sense explained below) numbers of such maxima in the sparse case. The basic idea of the upper bound proof is to consider a random assignment to some subset of the variables and determine the probability that some fraction of the remaining variables can be fixed without considering interactions between them. The bounded results hold in the case of weighted MAX-2-SAT as well. Using this technique and combining with ideas from Ref. 6, we find an algorithm for weighted MAX-2-SAT which is faster for large $d$ than previous algorithms which use polynomial space; this algorithm does require an additional bounds on maximum weights and degree.

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arXiv2016

Quantum Codes from High-Dimensional Manifolds

We construct toric codes on various high-dimensional manifolds. Assuming a conjecture in geometry we find families of quantum CSS stabilizer codes on $N$ qubits with logarithmic weight stabilizers and distance $N^{1-ε}$ for any $ε>0$. The conjecture is that there is a constant $C>0$ such that for any $n$-dimensional torus ${\mathbb T}^n={\mathbb R}^n/Λ$, where $Λ$ is a lattice, the least volume unoriented $n/2$-dimensional surface (using the Euclidean metric) representing nontrivial homology has volume at least $C^n$ times the volume of the least volume $n/2$-dimensional hyperplane representing nontrivial homology; in fact, it would suffice to have this result for $Λ$ an integral lattice with the surface restricted to faces of a cubulation by unit hypercubes. The main technical result is an estimate of Rankin invariants\cite{rankin} for certain random lattices, showing that in a certain sense they are optimal. Additionally, we construct codes with square-root distance, logarithmic weight stabilizers, and inverse polylogarithmic soundness factor (considered as quantum locally testable codes\cite{qltc}). We also provide an short, alternative proof that the shortest vector in the exterior power of a lattice may be non-split\cite{coulangeon}.

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arXiv2016

Random MERA States and the Tightness of the Brandao-Horodecki Entropy Bound

We construct a random MERA state with a bond dimension that varies with the level of the MERA. This causes the state to exhibit a very different entanglement structure from that usually seen in MERA, with neighboring intervals of length $l$ exhibiting a mutual information proportional to $εl$ for some constant $ε$, up to a length scale exponentially large in $ε$. We express the entropy of a random MERA in terms of sums over cuts through the MERA network, with the entropy in this case controlled by the cut minimizing bond dimensions cut through. One motivation for this construction is to investigate the tightness of the Brandao-Horodecki\cite{bh} entropy bound relating entanglement to correlation decay. Using the random MERA, we show that at least part of the proof is tight: there do exist states with the required property of having linear mutual information between neighboring intervals at all length scales. We conjecture that this state has exponential correlation decay and that it demonstrates that the Brandao-Horodecki bound is tight (at least up to constant factors), and we provide some numerical evidence for this as well as a sketch of how a proof of correlation decay might proceed.

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arXiv2016

Training A Quantum Optimizer

We study a variant of the quantum approximate optimization algorithm [ E. Farhi, J. Goldstone, and S. Gutmann, arXiv:1411.4028] with slightly different parametrization and different objective: rather than looking for a state which approximately solves an optimization problem, our goal is to find a quantum algorithm that, given an instance of MAX-2-SAT, will produce a state with high overlap with the optimal state. Using a machine learning approach, we chose a "training set" of instances and optimized the parameters to produce large overlap for the training set. We then tested these optimized parameters on a larger instance set. As a training set, we used a subset of the hard instances studied by E. Crosson, E. Farhi, C. Yen-Yu Lin, H.-H. Lin, and P. Shor (CFLLS) [arXiv:1401.7320]. When tested on the full set, the parameters that we find produce significantly larger overlap than the optimized annealing times of CFLLS. Testing on other random instances from $20$ to $28$ bits continues to show improvement over annealing, with the improvement being most notable on the hardest instances. Further tests on instances of MAX-3-SAT also showed improvement on the hardest instances. This algorithm may be a possible application for near-term quantum computers with limited coherence times.

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arXiv2016

The Asymptotics of Quantum Max-Flow Min-Cut

The quantum max-flow min-cut conjecture relates the rank of a tensor network to the minimum cut in the case that all tensors in the network are identical\cite{mfmc1}. This conjecture was shown to be false in Ref. \onlinecite{mfmc2} by an explicit counter-example. Here, we show that the conjecture is almost true, in that the ratio of the quantum max-flow to the quantum min-cut converges to $1$ as the dimension $N$ of the degrees of freedom on the edges of the network tends to infinity. The proof is based on estimating moments of the singular values of the network. We introduce a generalization of "rainbow diagrams"\cite{rainbow} to tensor networks to estimate the dominant diagrams. A direct comparison of second and fourth moments lower bounds the ratio of the quantum max-flow to the quantum min-cut by a constant. To show the tighter bound that the ratio tends to $1$, we consider higher moments. In addition, we show that the limiting moments as $N \rightarrow \infty$ agree with that in a different ensemble where tensors in the network are chosen independently, this is used to show that the distributions of singular values in the two different ensembles weakly converge to the same limiting distribution. We present also a numerical study of one particular tensor network, which shows a surprising dependence of the rank deficit on $N \mod 4$ and suggests further conjecture on the limiting behavior of the rank.

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arXiv2015

Towards Practical Quantum Variational Algorithms

The preparation of quantum states using short quantum circuits is one of the most promising near-term applications of small quantum computers, especially if the circuit is short enough and the fidelity of gates high enough that it can be executed without quantum error correction. Such quantum state preparation can be used in variational approaches, optimizing parameters in the circuit to minimize the energy of the constructed quantum state for a given problem Hamiltonian. For this purpose we propose a simple-to-implement class of quantum states motivated by adiabatic state preparation. We test its accuracy and determine the required circuit depth for a Hubbard model on ladders with up to 12 sites (24 spin-orbitals), and for small molecules. We find that this ansatz converges faster than previously proposed schemes based on unitary coupled clusters. While the required number of measurements is astronomically large for quantum chemistry applications to molecules, applying the variational approach to the Hubbard model (and related models) is found to be far less demanding and potentially practical on small quantum computers. We also discuss another application of quantum state preparation using short quantum circuits, to prepare trial ground states of models faster than using adiabatic state preparation.

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arXiv2015

Reduced Space-Time and Time Costs Using Dislocation Codes and Arbitrary Ancillas

We propose two distinct methods of improving quantum computing protocols based on surface codes. First, we analyze the use of dislocations instead of holes to produce logical qubits, potentially reducing spacetime volume required. Dislocations induce defects which, in many respects, behave like Majorana quasi-particles. We construct circuits to implement these codes and present fault-tolerant measurement methods for these and other defects which may reduce spatial overhead. One advantage of these codes is that Hadamard gates take exactly $0$ time to implement. We numerically study the performance of these codes using a minimum weight and a greedy decoder using finite-size scaling. Second, we consider state injection of arbitrary ancillas to produce arbitrary rotations. This avoids the logarithmic (in precision) overhead in online cost required if $T$ gates are used to synthesize arbitrary rotations. While this has been considered before, we consider also the parallel performance of this protocol. Arbitrary ancilla injection leads to a probabilistic protocol in which there is a constant chance of success on each round; we use an amortized analysis to show that even in a parallel setting this leads to only a constant factor slowdown as opposed to the logarithmic slowdown that might be expected naively.

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arXiv2014

Connecting Entanglement in Time and Space: Improving the Folding Algorithm

The "folding algorithm"\cite{fold1} is a matrix product state algorithm for simulating quantum systems that involves a spatial evolution of a matrix product state. Hence, the computational effort of this algorithm is controlled by the temporal entanglement. We show that this temporal entanglement is, in many cases, equal to the spatial entanglement of a modified Hamiltonian. This inspires a modification to the folding algorithm, that we call the "hybrid algorithm". We find that this leads to improved accuracy for the same numerical effort. We then use these algorithms to study relaxation in a transverse plus parallel field Ising model, finding persistent quasi-periodic oscillations for certain choices of initial conditions.

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arXiv2014

The Trotter Step Size Required for Accurate Quantum Simulation of Quantum Chemistry

The simulation of molecules is a widely anticipated application of quantum computers. However, recent studies \cite{WBCH13a,HWBT14a} have cast a shadow on this hope by revealing that the complexity in gate count of such simulations increases with the number of spin orbitals $N$ as $N^8$, which becomes prohibitive even for molecules of modest size $N\sim 100$. This study was partly based on a scaling analysis of the Trotter step required for an ensemble of random artificial molecules. Here, we revisit this analysis and find instead that the scaling is closer to $N^6$ in worst case for real model molecules we have studied, indicating that the random ensemble fails to accurately capture the statistical properties of real-world molecules. Actual scaling may be significantly better than this due to averaging effects. We then present an alternative simulation scheme and show that it can sometimes outperform existing schemes, but that this possibility depends crucially on the details of the simulated molecule. We obtain further improvements using a version of the coalescing scheme of \cite{WBCH13a}; this scheme is based on using different Trotter steps for different terms. The method we use to bound the complexity of simulating a given molecule is efficient, in contrast to the approach of \cite{WBCH13a,HWBT14a} which relied on exponentially costly classical exact simulation.

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arXiv2014

Notes on Some Questions in Mathematical Physics and Quantum Information

This is a set of notes on some unrelated topics in mathematical physics, at varying levels of detail. First, I consider certain questions relating to the decay of correlation functions in matrix product states, in particular those generated by quantum expanders. This is discussed in relation to recent results of Brandao and Horodecki on area laws on systems with exponentially decaying correlation function\cite{areaexp}. Second, I consider some difficulties in trying to construct a tensor product state (or PEPS) describing a two-dimensional fermionic system with non-vanishing Hall conductance. Third, I present some relations between the theory of almost commuting matrices and that of vector bundles, making the connection between the classifications more explicit. Fourth, I present an open question about quantum channels, and some partial results.

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arXiv2014

Improving Quantum Algorithms for Quantum Chemistry

We present several improvements to the standard Trotter-Suzuki based algorithms used in the simulation of quantum chemistry on a quantum computer. First, we modify how Jordan-Wigner transformations are implemented to reduce their cost from linear or logarithmic in the number of orbitals to a constant. Our modification does not require additional ancilla qubits. Then, we demonstrate how many operations can be parallelized, leading to a further linear decrease in the parallel depth of the circuit, at the cost of a small constant factor increase in number of qubits required. Thirdly, we modify the term order in the Trotter-Suzuki decomposition, significantly reducing the error at given Trotter-Suzuki timestep. A final improvement modifies the Hamiltonian to reduce errors introduced by the non-zero Trotter-Suzuki timestep. All of these techniques are validated using numerical simulation and detailed gate counts are given for realistic molecules.

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arXiv2013

Matrix Product Operators and Central Elements: Classical Description of a Quantum State

We study planar two-dimensional quantum systems on a lattice whose Hamiltonian is a sum of local commuting projectors of bounded range. We consider whether or not such a system has a zero energy ground state. To do this, we consider the problem as a one-dimensional problem, grouping all sites along a column into "supersites"; using $C^*$-algebraic methods (Bravyi and Vyalyi), we can solve this problem if we can characterize the central elements of the interaction algebra on these supersite. Unfortunately, these central elements may be very complex, making brute force impractical. Instead, we show a characterization of these elements in terms of matrix product operators with bounded bond dimension. This bound can be interpreted as a bound on the number of particle types in lattice theories with bounded Hilbert space dimension on each site. Topological order in this approach is related to the existence of certain central elements which cannot be "broken" into smaller pieces without creating an end excitation. Using this bound on bond dimension, we prove that several special cases of this problem are in NP, and we give part of a proof that the general case is in NP. Further, we characterize central elements that appear in certain specific models, including toric code and Levin-Wen models, as either product operators in the Abelian case or matrix product operators with low bond dimension in the non-Abelian case; this matrix product operator representation may have practical application in engineering the complicated multi-spin interactions in the Levin-Wen models.

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