Deterministic transformations between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality

• URL: http://arxiv.org/abs/2109.08202v3
• Date: Mon, 28 Mar 2022 10:36:36 GMT
• Title: Deterministic transformations between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality
• Authors: Marco T\'ulio Quintino, Daniel Ebler
• Abstract summary: We show that when $f$ is an anti-homomorphism, sequential circuits could exponentially outperform parallel ones. We present explicit constructions on how to obtain such advantages for the unitary inversion task $f(U)=U-1$ and the unitary transposition task $f(U)=UT$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This work analyses the performance of quantum circuits and general processes to transform $k$ uses of an arbitrary unitary operation $U$ into another unitary operation $f(U)$. When the desired function $f$ a homomorphism, i.e., $f(UV)=f(U)f(V)$, it is known that optimal average fidelity is attainable by parallel circuits and indefinite causality does not provide any advantage. Here we show that the situation changes dramatically when considering anti-homomorphisms, i.e., $f(UV)=f(V)f(U)$. In particular, we prove that when $f$ is an anti-homomorphism, sequential circuits could exponentially outperform parallel ones and processes with indefinite causal order could outperform sequential ones. We presented explicit constructions on how to obtain such advantages for the unitary inversion task $f(U)=U^{-1}$ and the unitary transposition task $f(U)=U^T$. We also stablish a one-to-one connection between the problem of unitary estimation and parallel unitary transposition, allowing one to easily translate results from one field to the other. Finally, we apply our results to several concrete problem instances and present a method based on computer-assisted proofs to show optimality.

Related papers

• IT$^3$: Idempotent Test-Time Training [95.78053599609044]
  This paper introduces Idempotent Test-Time Training (IT$3$), a novel approach to addressing the challenge of distribution shift. IT$3$ is based on the universal property of idempotence. We demonstrate the versatility of our approach across various tasks, including corrupted image classification.
  arXiv  Detail & Related papers  (2024-10-05T15:39:51Z)
• Cubic power functions with optimal second-order differential uniformity [0.0]
  We prove that $d=22k+2k+1$ and $gcd(k,n)=1$ have optimal second-order differential uniformity. Up to affine equivalence, these might be the only optimal cubic power functions.
  arXiv  Detail & Related papers  (2024-09-05T12:22:32Z)
• Transfer Operators from Batches of Unpaired Points via Entropic Transport Kernels [3.099885205621181]
  We derive a maximum-likelihood inference functional, propose a computationally tractable approximation and analyze their properties. We prove a $Gamma$-convergence result showing that we can recover the true density from empirical approximations as the number $N$ of blocks goes to infinity.
  arXiv  Detail & Related papers  (2024-02-13T12:52:41Z)
• Transformers as Support Vector Machines [54.642793677472724]
  We establish a formal equivalence between the optimization geometry of self-attention and a hard-margin SVM problem. We characterize the implicit bias of 1-layer transformers optimized with gradient descent. We believe these findings inspire the interpretation of transformers as a hierarchy of SVMs that separates and selects optimal tokens.
  arXiv  Detail & Related papers  (2023-08-31T17:57:50Z)
• Optimal universal quantum circuits for unitary complex conjugation [1.6492989697868894]
  This work presents optimal quantum circuits for transforming a number $k$ of calls of $U_d$ into its complex conjugate $barU_d$. Our circuits admit a parallel implementation and are proven to be optimal for any $k$ and $d$ with an average fidelity of $leftlangleFrightrangle =frack+1d(d-k)$.
  arXiv  Detail & Related papers  (2022-05-31T20:43:29Z)
• Resolving mean-field solutions of dissipative phase transitions using permutational symmetry [0.0]
  Phase transitions in dissipative quantum systems have been investigated using various analytical approaches, particularly in the mean-field (MF) limit. These two solutions cannot be reconciled because the MF solutions above $d_c$ should be identical. numerical studies on large systems may not be feasible because of the exponential increase in computational complexity.
  arXiv  Detail & Related papers  (2021-10-18T16:07:09Z)
• Feature Cross Search via Submodular Optimization [58.15569071608769]
  We study feature cross search as a fundamental primitive in feature engineering. We show that there exists a simple greedy $(1-1/e)$-approximation algorithm for this problem.
  arXiv  Detail & Related papers  (2021-07-05T16:58:31Z)
• Submodular + Concave [53.208470310734825]
  It has been well established that first order optimization methods can converge to the maximal objective value of concave functions. In this work, we initiate the determinant of the smooth functions convex body $$F(x) = G(x) +C(x)$. This class of functions is an extension of both concave and continuous DR-submodular functions for which no guarantee is known.
  arXiv  Detail & Related papers  (2021-06-09T01:59:55Z)
• Linear Time Sinkhorn Divergences using Positive Features [51.50788603386766]
  Solving optimal transport with an entropic regularization requires computing a $ntimes n$ kernel matrix that is repeatedly applied to a vector. We propose to use instead ground costs of the form $c(x,y)=-logdotpvarphi(x)varphi(y)$ where $varphi$ is a map from the ground space onto the positive orthant $RRr_+$, with $rll n$.
  arXiv  Detail & Related papers  (2020-06-12T10:21:40Z)
• $O(n)$ Connections are Expressive Enough: Universal Approximability of Sparse Transformers [71.31712741938837]
  We show that sparse Transformers with only $O(n)$ connections per attention layer can approximate the same function class as the dense model with $n2$ connections. We also present experiments comparing different patterns/levels of sparsity on standard NLP tasks.
  arXiv  Detail & Related papers  (2020-06-08T18:30:12Z)
• On Negative Transfer and Structure of Latent Functions in Multi-output Gaussian Processes [2.538209532048867]
  In this article, we first define negative transfer in the context of an $mathcalMGP$ and then derive necessary conditions for an $mathcalMGP$ model to avoid negative transfer. We show that avoiding negative transfer is mainly dependent on having a sufficient number of latent functions $Q$. We propose two latent structures that scale to arbitrarily large datasets, can avoid negative transfer and allow any kernel or sparse approximations to be used within.
  arXiv  Detail & Related papers  (2020-04-06T02:47:30Z)

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.