Improved Strong Simulation of Universal Quantum Circuits

• URL: http://arxiv.org/abs/2012.11739v4
• Date: Tue, 7 Jun 2022 00:20:55 GMT
• Title: Improved Strong Simulation of Universal Quantum Circuits
• Authors: Lucas Kocia
• Abstract summary: We find a scaling reduction in the stabilizer rank of the twelve-qubit tensored $T$ gate magic state. This lowers its bound to $2sim 0.463 t$ for multi-Pauli measurements on $t$ magic states. This constructively produces the most efficient strong simulation of the Clifford+$T$ gateset.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We find a scaling reduction in the stabilizer rank of the twelve-qubit tensored $T$ gate magic state. This lowers its asymptotic bound to $2^{\sim 0.463 t}$ for multi-Pauli measurements on $t$ magic states, improving over the best previously found bound of $2^{\sim 0.468 t}$. We numerically demonstrate this reduction. This constructively produces the most efficient strong simulation algorithm of the Clifford+$T$ gateset to relative or multiplicative error. We then examine the cost of Pauli measurement in terms of its Gauss sum rank, which is a slight generalization of the stabilizer rank and is a lower bound on its asymptotic scaling. We demonstrate that this lower bound appears to be tight at low $t$-counts, which suggests that the stabilizer rank found at the twelve-qubit state can be lowered further to $2^{\sim 0.449 t}$ and we prove and numerically show that this is the case for single-Pauli measurements. Our construction directly shows how the reduction at $12$ qubits is iteratively based on the reduction obtained at $6$, $3$, $2$, and $1$ qubits. This explains why novel reductions are found at tensor factors for these number of qubit primitives, an explanation lacking previously in the literature. Furthermore, in the process we observe an interesting relationship between the T gate magic state's stabilizer rank and decompositions that are Clifford-isomorphic to a computational sub-basis tensored with single-qubit states that produce minimal unique stabilizer state inner products -- the same relationship that allowed for finding minimal numbers of unique Gauss sums in the odd-dimensional qudit Wigner formulation of Pauli measurements.

Related papers

• Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path [80.60592344361073]
  We study the Shortest Path (SSP) problem with a linear mixture transition kernel. An agent repeatedly interacts with a environment and seeks to reach certain goal state while minimizing the cumulative cost. Existing works often assume a strictly positive lower bound of the iteration cost function or an upper bound of the expected length for the optimal policy.
  arXiv  Detail & Related papers  (2024-02-14T07:52:00Z)
• Towards large-scale quantum optimization solvers with few qubits [59.63282173947468]
  We introduce a variational quantum solver for optimizations over $m=mathcalO(nk)$ binary variables using only $n$ qubits, with tunable $k>1$. We analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature.
  arXiv  Detail & Related papers  (2024-01-17T18:59:38Z)
• Quadratic Lower bounds on the Approximate Stabilizer Rank: A Probabilistic Approach [6.169364905804677]
  The approximate stabilizer rank of a quantum state is the minimum number of terms in any approximate decomposition of that state into stabilizer states. We improve the lower bound on the approximate rank to $tilde Omega(sqrt n)$ for a wide range of the approximation parameters.
  arXiv  Detail & Related papers  (2023-05-17T15:09:41Z)
• Causal Bandits for Linear Structural Equation Models [58.2875460517691]
  This paper studies the problem of designing an optimal sequence of interventions in a causal graphical model. It is assumed that the graph's structure is known and has $N$ nodes. Two algorithms are proposed for the frequentist (UCB-based) and Bayesian settings.
  arXiv  Detail & Related papers  (2022-08-26T16:21:31Z)
• Quantifying nonstabilizerness of matrix product states [0.0]
  We show that nonstabilizerness, as quantified by the recently introduced Stabilizer R'enyi Entropies (SREs), can be computed efficiently for matrix product states (MPSs) We exploit this observation to revisit the study of ground-state nonstabilizerness in the quantum Ising chain, providing accurate numerical results up to large system sizes.
  arXiv  Detail & Related papers  (2022-07-26T17:50:32Z)
• Improved Graph Formalism for Quantum Circuit Simulation [77.34726150561087]
  We show how to efficiently simplify stabilizer states to canonical form. We characterize all linearly dependent triplets, revealing symmetries in the inner products. Using our novel controlled-Pauli $Z$ algorithm, we improve runtime for inner product computation from $O(n3)$ to $O(nd2)$ where $d$ is the maximum degree of the graph.
  arXiv  Detail & Related papers  (2021-09-20T05:56:25Z)
• Lower Bounds on Stabilizer Rank [3.265773263570237]
  We prove that for a sufficiently small constant $delta, the stabilizer rank of any state which is $$-close to those states is $Omega(sqrtn/log n)$. This is the first non-trivial lower bound for approximate stabilizer rank.
  arXiv  Detail & Related papers  (2021-06-06T19:27:51Z)
• Improved Sample Complexity for Incremental Autonomous Exploration in MDPs [132.88757893161699]
  We learn the set of $epsilon$-optimal goal-conditioned policies attaining all states that are incrementally reachable within $L$ steps. DisCo is the first algorithm that can return an $epsilon/c_min$-optimal policy for any cost-sensitive shortest-path problem.
  arXiv  Detail & Related papers  (2020-12-29T14:06:09Z)
• Early fault-tolerant simulations of the Hubbard model [3.988614978933934]
  Simulation of the Hubbard model is a leading candidate for the first useful applications of a fault-tolerant quantum computer. We present a new analytic approach to bounding the simulation error due to Trotterization that provides much tighter bounds for the split-operator FFFT method. We find there is a potentially useful application for fault-tolerant quantum computers using around one million Toffoli gates.
  arXiv  Detail & Related papers  (2020-12-16T20:07:54Z)
• Improved Simulation of Quantum Circuits by Fewer Gaussian Eliminations [0.0]
  We show that the cost of strong simulation of quantum circuits using $t$ $T$ gate magic states exhibits non-trivial reductions on its upper bound. This agrees with previous numerical bounds found for qubits.
  arXiv  Detail & Related papers  (2020-03-02T19:00:04Z)
• Quantum Algorithms for Simulating the Lattice Schwinger Model [63.18141027763459]
  We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In lattice units, we find a Schwinger model on $N/2$ physical sites with coupling constant $x-1/2$ and electric field cutoff $x-1/2Lambda$. We estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable---the mean pair density.
  arXiv  Detail & Related papers  (2020-02-25T19:18:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.