Abstract Currently available quantum computers, so called Noisy Intermediate-Scale Quantum devices, are characterized by relatively low number of qubits and moderate gate fidelities. In such scenario, the implementation of quantum error correction is impossible and the performance of those devices is quite modest. In particular, the depth of circuits implementable with reasonably high fidelity is limited, and the minimization of circuit depth is required. Such depths depend on the efficiency of the universal set of gates used in computation, and can be bounded using the Solovay–Kitaev theorem. However, it is known that much better, asymptotically tight bounds of the form , can be obtained for specific . Those bounds are controlled by so called spectral gap, denoted . Yet, the computation of is not possible for general and in practice one considers spectral gap at a certain scale r ( ϵ ) , denoted . This turns out to be sufficient to bound the efficiency of provided that one is interested in a physically feasible case, in which an error ϵ is bounded from below. In this paper we derive lower bounds on and, as a consequence, on the efficiency of universal sets of d-dimensional quantum gates satisfying an additional condition. The condition is naturally met for generic quantum gates, such as e.g. Haar random gates. Our bounds are explicit in the sense that all parameters can be determined by numerical calculations on existing computers, at least for small d. This is in contrast with known lower bounds on which involve parameters with ambiguous values.
Abstract Currently available quantum computers, so called Noisy Intermediate-Scale Quantum devices, are characterized by relatively low number of qubits and moderate gate fidelities. In such scenario, the implementation of quantum error correction is impossible and the performance of those devices is quite modest. In particular, the depth of circuits implementable with reasonably high fidelity is limited, and the minimization of circuit depth is required. Such depths depend on the efficiency of the universal set of gates used in computation, and can be bounded using the Solovay–Kitaev theorem. However, it is known that much better, asymptotically tight bounds of the form , can be obtained for specific . Those bounds are controlled by so called spectral gap, denoted . Yet, the computation of is not possible for general and in practice one considers spectral gap at a certain scale r ( ϵ ) , denoted . This turns out to be sufficient to bound the efficiency of provided that one is interested in a physically feasible case, in which an error ϵ is bounded from below. In this paper we derive lower bounds on and, as a consequence, on the efficiency of universal sets of d-dimensional quantum gates satisfying an additional condition. The condition is naturally met for generic quantum gates, such as e.g. Haar random gates. Our bounds are explicit in the sense that all parameters can be determined by numerical calculations on existing computers, at least for small d. This is in contrast with known lower bounds on which involve parameters with ambiguous values.
Quantum computation is of current ubiquitous interest in physics, computer science, and the public interest. In the not-so-distant future, quantum computers will be relatively common pieces of research equipment. Eventually, one can expect an actively quantum computer to be a common feature of life. In this work, I study the approximation efficiency of several common universal quantum gate sets at short sequence lengths using an implementation of the Solovay-Kitaev algorithm. I begin by developing from almost nothing the relevant formal mathematics to rigorously describe what one means by the terms universal gate set and covering efficiency. I then describe some interesting results on the asymptotic covering properties of certain classes of universal gate sets and discuss the theorem which the Solovay-Kitaev algorithm is based on. Moving from mathematical introduction to experimental method, I then describe how sets will be compared. I use the commonly studied sets H+T, Pauli+V, V, and Clifford+T to determine which is the most efficient at approximating randomly generated unitaries. By doing so, we get an understanding of how well each set would perform in the context of a general quantum computer processor. This was accomplished by using the same implementation of the Solovay-Kitaev algorithm throughout, with roughly equal-sized preprocessed libraries formed from each gate set, over approximations for 10,000 randomly generated unitary matrices at algorithm depth n=5. Ultimately, the Pauli+V and V sets were the most efficient and had similar performance qualities. On average the Pauli+V set produced approximations of length 15,491 and accuracy 0.0002686. The V basis produced approximations of average sequence length 16,403 and accuracy 0.0001465. This performance is about equal given this particular implementation of the Solovay-Kitaev algorithm. We conclude that this result is somewhat surprising as the general behavior and efficiency of these particular choices of gate set are expected to be similar. It is possible though that the asymptotic efficiencies of these gate sets vary by a relatively wide margin and this has effected the experiment. It is also possible that some aspect of a naive implementation of the Solovay-Kitaev algorithm resulted in the Hadamard gate based sets performing more poorly than the V basis sets overall. Due to constraints on computational power, this result could also be limited to this particular accuracy regime and could even out as tolerance 𝜀 is taken to be arbitrarily small. Further possibilities of this result as well as further work are then discussed.
We are rapidly reaching the point with today's quantum computers where we will be unable to verify their outcomes with our current classical computers.Arguably, in some applications, this threshold has already been reached [1].Soon this may also be the case with fault-tolerant or near-term intermediate-scale quantum (NISQ) computers [2] that implement useful quantum algorithms.Validating the performance of NISQ devices and establishing a clearly-dened threshold for when they can outperform current computers is accomplished by directly simulating their outcomes on classical computers.The community needs more ecient classical algorithms and they need to know how much more ecient they can get.This last desired quality is often missing in currently available methods.Here, we accomplish both of these goals.Universal quantum computation requires a universal gateset, appropriate initial states and set of measurements.A common universal set that is widely used consists of stabilizer states, the Cliord+T gateset, and Pauli measurements.For classical computers, simulating everything other than the T gates can be done very eciently in polynomial time [3].On the other hand, T gates have been found to be exponentially dicult to simulate.Fortunately, classical simulation of universal quantum outcomes is made easier somewhat by the fact that quantum computers only produce output up to additive error [4], and so there is a priori no reason to simulate their outcomes beyond this.Simulating a given state as it undergoes Cliord and T gate unitary evolution up to some additive error is an L 1 minimization problem, which are known to scale more favorably compared to L 0 or L 2 minimizations appropriate when there are more stringent error constraints.The most natural state decomposition that is used for such classical simulation is the stabilizer state decomposition, because classical computers can calculate their inner products especially eciently.The corresponding number of these stabilizer states sucient for additive error decompositions is called the stabilizer extent.Current state-of-the-art classical simulators of quantum outcomes from this gateset scale as O(ξ t δ -2 ) [57] and O(ξ t δ -1 ) [8], where ξ t ≡ 2 ∼0.228t is the stabilizer extent of the T gate magic state and δ is the additive error.Furthermore, it is conjectured [6] that this asymptotic scaling is tight w.r.t.t.The most promising route to nd reductions in this algorithm is to look for at the next-order terms.f t : Reduction Relative Reduction Compared to i.d.d.Sampling w.r.t.t t FIG. 1. [From Technical Paper #1] The relative improvement in scaling w.r.t.t produced by correlated sampling with dierent 0 ≤ ft ≤ O(ξ 2t/3 ) compared to the state-of-the-art.In practice, the desired δ determines which ft you can attain:Despite their dimunitive name, next-order eects can oer signicant improvements for the intermediate (nonasymptotic) t regime, which is precisely the regime that NISQ devices are in.For a sense of this, see Figure 1 for a preview of the relative improvement in scaling that can be attained from including such next-order terms.Unfortunately, existing methods all saturate the asymptotic conjectured lower bound w.r.t.t even when they are not in the asymptotic limit of t → ∞ and it is not immediately clear how to extend them to next order in t.We take a rst crack at this problem in [9] (attached Technical Paper #2) and introduce the tool of correlated L 1 sampling.The main technique that we use is supplementing independent L 1 sampling with correlated samples of dissimilar states that lead to a better average.Given an ensemble of M independently sampled states, which have an SAND2021-13613CThis paper describes objective technical results and analysis.Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S.
An ultimate goal of quantum computing is to perform calculations beyond the reach of any classical computer.It is therefore imperative that useful quantum computers be very difficult to simulate classically, otherwise classical computers could be used for the applications envisioned for the quantum ones.Perfect quantum computers are unarguably exponentially difficult to simulate: the classical resources required grow exponentially with the number of qubits N or the depth D of the circuit.This difficulty has triggered recent experiments on deep, random circuits that aim to demonstrate that quantum devices may already perform tasks beyond the reach of classical computing.These real quantum computing devices, however, suffer from many sources of decoherence and imprecision which limit the degree of entanglement that can actually be reached to a fraction of its theoretical maximum.They are characterized by an exponentially decaying fidelity F ∼ ð1 -ϵÞ ND with an error rate ϵ per operation as small as ≈1% for current devices with several dozen qubits or even smaller for smaller devices.In this work, we provide new insight on the computing capabilities of real quantum computers by demonstrating that they can be simulated at a tiny fraction of the cost that would be needed for a perfect quantum computer.Our algorithms compress the representations of quantum wave functions using matrix product states, which are able to capture states with low to moderate entanglement very accurately.This compression introduces a finite error rate ϵ so that the algorithms closely mimic the behavior of real quantum computing devices.The computing time of our algorithm increases only linearly with N and D in sharp contrast with exact simulation algorithms.We illustrate our algorithms with simulations of random circuits for qubits connected in both one-and twodimensional lattices.We find that ϵ can be decreased at a polynomial cost in computing power down to a minimum error ϵ ∞ .Getting below ϵ ∞ requires computing resources that increase exponentially with ϵ ∞ =ϵ.For a two-dimensional array of N ¼ 54 qubits and a circuit with control-Z gates, error rates better than state-of-the-art devices can be obtained on a laptop in a few hours.For more complex gates such as a SWAP gate followed by a controlled rotation, the error rate increases by a factor 3 for similar computing time.Our results suggest that, despite the high fidelity reached by quantum devices, only a tiny fraction ð∼10 -8 Þ of the system Hilbert space is actually being exploited.
A fundamental question is understanding the rate at which random quantum circuits converge to the Haar measure. One quantity which is important in establishing this rate is the spectral gap of a random quantum ensemble. In this work we establish a new bound on the spectral gap of the t-th moment of a one-dimensional brickwork architecture on N qudits. This bound is independent of both t and N, provided t does not exceed the qudit dimension q. We also show that the bound is nearly optimal. The improved spectral gaps gives large improvements to the constant factors in known results on the approximate t-design depths of the 1D brickwork, of generic circuit architectures, and of specially-constructed architectures which scramble in depth O(log N). We moreover show that the spectral gap gives the dominant epsilon-dependence of the t-design depth at small epsilon. Our spectral gap bound is obtained by bounding the N-site 1D brickwork architecture by the spectra of 3-site operators. We then exploit a block-triangular hierarchy and a global symmetry in these operators in order to efficiently bound them. The technical methods used are a qualitatively different approach for bounding spectral gaps and and have little in common with previous techniques.
The spectral gap of local random quantum circuits is a fundamental property that determines how close the moments of the circuit's unitaries match those of a Haar random distribution. When studying spectral gaps, it is common to bound these quantities using tools from statistical mechanics or via quantum information-based inequalities. By focusing on the second moment of one-dimensional unitary circuits where nearest-neighboring gates act on sets of qudits (with open and closed boundary conditions), we show that one can exactly compute the associated spectral gaps. Indeed, having access to their functional form allows us to prove several important results, such as the fact that the spectral gap for closed boundary condition is exactly the square of the gap for open boundaries, as well as improve on previously known bounds for approximate design convergence. Finally, we verify our theoretical results by numerically computing the spectral gap for systems of up to 70 qubits, as well as comparing them to gaps of random orthogonal and symplectic circuits.
Random quantum circuits are a central concept in quantum information theory with applications ranging from demonstrations of quantum computational advantage to descriptions of scrambling in strongly interacting systems and black holes. The utility of random quantum circuits in these settings stems from their ability to rapidly generate quantum pseudorandomness. In a seminal paper by Brand\~ao, Harrow, and Horodecki [Commun. Math. Phys. 346, 397 (2016)] it was proven that the tth moment operator of local random quantum circuits on n qudits with local dimension q has a spectral gap of at least $\mathrm{\ensuremath{\Omega}}({n}^{\ensuremath{-}1}{t}^{\ensuremath{-}5\ensuremath{-}3.1/ln(q)})$, which implies that they are efficient constructions of approximate unitary designs. As a first result, we use Knabe bounds for the spectral gaps of frustration-free Hamiltonians to show that one-dimensional random quantum circuits have a spectral gap scaling as $\mathrm{\ensuremath{\Omega}}({n}^{\ensuremath{-}1})$, provided that t is small compared to the local dimension: ${t}^{2}\ensuremath{\le}O(q)$. This implies a (nearly) linear scaling of the circuit depth in the design order t. Our second result is an unconditional spectral gap bounded below by $\mathrm{\ensuremath{\Omega}}[{n}^{\ensuremath{-}1}{ln}^{\ensuremath{-}1}(n){t}^{\ensuremath{-}\ensuremath{\alpha}(q)}]$ for random quantum circuits with all-to-all interactions. This improves both the n and t scaling in design depth for the nonlocal model. We show this by proving a recursion relation for the spectral gaps involving an auxiliary random walk. Lastly, we solve the smallest nontrivial case exactly and combine with numerics and Knabe bounds to improve the constants involved in the spectral gap for small values of t.