Given two dense multivectors of the Clifford algebra $C\ell(V, Q)$ with $N=2^{p+q}$ coefficients, the fastest known classical algorithms compute their geometric product in $O(N^{ω/2})$ arithmetic operations, where $ω$ denotes the matrix multiplication exponent. I show that, under amplitude encoding, a quantum computer executes the geometric product in $O(\operatorname{polylog} N)$ time, using logarithmic space with sublogarithmic circuit depth. This exponential speedup establishes Clifford multiplication as a quantum primitive, providing an efficient computational foundation for quantum geometric algorithms and relativistic simulations.
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The paper asks whether the geometric product of two Clifford multivectors can be computed efficiently from amplitude-encoded coefficient states. For an algebra with $N=2^n$ basis blades, it expresses multiplication as cocycle-twisted convolution over $(\mathbb Z_2)^n$: $c_z=\sum_x a_xb_{x\oplus z}\chi(x,x\oplus z)$. The proposed circuit applies the cocycle phase, computes XOR between the registers, and applies a Walsh–Hadamard transform to the first register. Postselecting that register onto $0^n$ produces the normalized coefficient state of the product.
The cocycle is a quadratic Boolean phase encoding generator exchanges and metric signs. A reversible binary prefix-sum construction implements its oracle using $O(n)$ two-qubit gates and depth $O(n)$; a parallel-prefix variant is stated to use $O(n\log n)$ gates and depth $O(\log n)$. These give corresponding per-attempt multiplication costs, with $O(n)$ qubits. The construction is also formulated for efficiently presented twisted group algebras.
The central qualification is the success probability: for normalized inputs, $p_0=\|AB\|_2^2/N$. The paper estimates that typical random dense inputs have $p_0=O(1/N)$, so the small per-attempt circuit does not establish generic polylogarithmic-time product-state preparation. Amplitude amplification requires $O(1/\sqrt{p_0})$ iterations and efficient input-state reflections. Support-adapted projection can improve success for small supports, although sparse multiplication is also classically efficient. Arbitrary input preparation and full classical output extraction require resources scaling with $N$. Proposed applications concern product-state observables under efficient preparation and inverse-polynomial-success promises; no concrete application speedup is established.
Some signs of AI-generated textOverclaimsUnclear noveltyStrong assumptionsMissing comparison
Central claim may not hold: The abstract claims general polylogarithmic-time multiplication, but the circuit succeeds with probability ||AB||_2^2/N, which can entail exponentially many repetitions in the geometric dimension.
Efficient product-state preparation requires additional success-probability and state-preparation promises that amplitude encoding alone does not provide.
The advertised sublogarithmic depth applies to a single attempt rather than to successful product-state preparation with bounded failure probability.
The exponential-speedup claim compares classical full-array computation with conditional quantum-state preparation without establishing an exponential advantage for a matched computational task.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
Given two dense multivectors of the Clifford algebra Cℓ(V,Q) with N=2p+q coefficients, the fastest known classical algorithms compute their geometric product in O(Nω/2) arithmetic operations, where ω denotes the matrix multiplication exponent. I show that, under amplitude encoding, a quantum computer executes the geometric product in O(polylogN) time, using logarithmic space with sublogarithmic circuit depth. This exponential speedup establishes Clifford multiplication as a quantum primitive, providing an efficient computational foundation for quantum geometric algorithms and relativistic simulations.
The paper asks whether the geometric product of two Clifford multivectors can be computed efficiently from amplitude-encoded coefficient states. For an algebra with N=2n basis blades, it expresses multiplication as cocycle-twisted convolution over (Z2)n: cz=∑xaxbx⊕zχ(x,x⊕z). The proposed circuit applies the cocycle phase, computes XOR between the registers, and applies a Walsh–Hadamard transform to the first register. Postselecting that register onto 0n produces the normalized coefficient state of the product.
The cocycle is a quadratic Boolean phase encoding generator exchanges and metric signs. A reversible binary prefix-sum construction implements its oracle using O(n) two-qubit gates and depth O(n); a parallel-prefix variant is stated to use O(nlogn) gates and depth O(logn). These give corresponding per-attempt multiplication costs, with O(n) qubits. The construction is also formulated for efficiently presented twisted group algebras.
The central qualification is the success probability: for normalized inputs, p0=∥AB∥22/N. The paper estimates that typical random dense inputs have p0=O(1/N), so the small per-attempt circuit does not establish generic polylogarithmic-time product-state preparation. Amplitude amplification requires O(1/p0) iterations and efficient input-state reflections. Support-adapted projection can improve success for small supports, although sparse multiplication is also classically efficient. Arbitrary input preparation and full classical output extraction require resources scaling with N. Proposed applications concern product-state observables under efficient preparation and inverse-polynomial-success promises; no concrete application speedup is established.
This article presents two quantum algorithms for computing the product of a circulant matrix and a vector. The arithmetic complexity of the first algorithm is O(Nlog2N) in most cases. For the second algorithm, when the entries in the circulant matrix and the vector take values in C or R, the complexity is O(Nlog2N) in most cases. However, when these entries take values from positive real numbers, the complexity is reduced to O(log3N) in most cases, which presents an exponential speedup compared to the classical complexity of O(NlogN) for computing the product of a circulant matrix and vector. We apply this algorithm to the convolution calculation in quantum convolutional neural networks, which effectively accelerates the computation of convolutions. Additionally, we present a concrete quantum circuit structure for quantum convolutional neural networks.
Random Clifford operators have numerous applications in quantum computing, including randomized benchmarking, classical shadows, and quantum authentication. However, sampling and implementing uniformly random n-qubit Clifford incur near-quadratic complexity due to the size of Clifford group.
We introduce a cryptographic way to overcome these barriers: trapdoored Clifford operator distributions whose samples are computationally indistinguishable from uniformly random Cliffords, yet implementing them can be much faster given the trapdoor. We construct a distribution of trapdoored Clifford operators whose elements can be sampled and implemented in near-linear time under a variant of the learning parity with noise assumption. Our constructions allow fast tableau action on Pauli labels for classical simulation, and also can be optimized to admit polylogarithmic-depth implementation. Along the way, we construct trapdoored matrices over finite fields that support efficient multiplication by both a matrix and its inverse, resolving an open question left by Vaikuntanathan and Zamir [SODA'26].
We use these constructions to obtain faster protocols based on random Cliffords. We also explore their applications to the worst-case to average-case reductions for matrix and Clifford problems including the iterated matrix multiplication and Clifford circuit synthesis. In particular, we show the hardness of batching Clifford circuits: synthesizing circuits that apply the same Clifford to multiple registers is at least as hard as worst-case matrix multiplication, even when synthesis succeeds on a small constant fraction of random Cliffords. This extends to approximate implementations by general quantum circuits.
This paper presents a unified algebraic framework for n-qubit quantum computation rooted in the complex Clifford algebra Cl(2n,C). We leverage the fundamental isomorphism Cl(2n,C)≅M(2n,C) to directly identify the algebra of all n-qubit quantum operators (including states, gates, and observables) with elements of Cl(2n,C). This operator-centric perspective treats quantum states, whether pure or mixed, as density multivectors ρ∈Cl(2n,C) satisfying standard physical constraints (ρ†=ρ, 2n⟨ρ⟩0=1, positivity), and unitary gates U as multivectors obeying U†U=I. Quantum evolution ρ′=UρU† and measurements p(m)=2n⟨Pmρ⟩0 are thus described by intrinsic Clifford algebraic operations. This contrasts with, yet encompasses, specific realizations like spinor representations based on Witt ideals (S=Cl(2n,C)∏jfjfj†), which are shown to be particular constructions within this universal algebra. By analyzing Deutsch's algorithm through these different lenses, we demonstrate the generality and unifying power of the operator-centric multivector approach. This formulation not only provides a comprehensive algebraic language that naturally handles mixed states and general quantum processes but also opens avenues for applying the rich geometric and structural tools of Clifford algebras—such as grade decomposition and invariant theory—to gain deeper insights into quantum information, entanglement, and dynamics.
Quantum computing has potential to provide exponential speedups over classical computing for many important applications. However, today's quantum computers are in their early stages, and hardware quality issues hinder the scale of program execution. Benchmarking and simulation of quantum circuits on classical computers is therefore essential to advance the understanding of how quantum computers and programs operate, enabling both algorithm discovery that leads to high-impact quantum computation and engineering improvements that deliver to more powerful quantum systems. Unfortunately, the nature of quantum information causes simulation complexity to scale exponentially with problem size.
Quantum algorithms for computational linear algebra promise up to exponential speedups for applications such as simulation and regression, making them prime candidates for hardware realization. But these algorithms execute in a model that cannot efficiently store matrices in memory like a classical algorithm does, instead requiring developers to implement complex expressions for matrix arithmetic in terms of correct and efficient quantum circuits. Among the challenges for the developer is navigating a cost model in which conventional optimizations for linear algebra, such as subexpression reuse, can be inapplicable or unprofitable.
In this work, we present Cobble, a language for programming with quantum computational linear algebra. Cobble enables developers to express and manipulate the quantum representations of matrices, known as block encodings, using high-level notation that automatically compiles to correct quantum circuits. Cobble features analyses that compute the time and space usage of programs, as well as optimizations that reduce overhead and generate efficient circuits using state-of-the-art techniques such as the quantum singular value transformation. We evaluate Cobble on benchmark kernels for simulation, regression, search, and other applications, showing 2.6x-25.4x speedups on these benchmarks compared to the unoptimized baseline.
Despite the widespread belief that quantum computers cannot be efficiently simulated classically, efficient simulation is known to be possible in certain restricted regimes. In particular, the Gottesman-Knill theorem states that Clifford circuits can be efficiently simulated. We begin this thesis by reviewing and comparing several known techniques for efficient simulation of Clifford circuits: the stabilizer formalism, CH form, affine form, and the graph state formalism. We describe each simulation method and give four different proofs of the Gottesman-Knill theorem. \n \nNext we review a recent work [15], which shows that restricting the geometry of Clifford circuits can lead to a further speedup. We give an algorithm for simulating Pauli basis measurements on a planar graph state in time widetildeO(nomega/2), where omega<2.373 is the matrix multiplication exponent. This algorithm achieves a quadratic speedup over using Clifford simulation methods directly. As an application of this algorithm, we consider a depth-d Clifford circuit whose two-qubit gates act along edges of a planar graph and describe how to sample from its output distribution or compute an output probability in time $\\widetilde{O}(n^{\\omega/2}d^\\omega)$. For d=O(logn), both of these results are quadratic speedups over using Clifford simulation methods directly. \n \nFinally, we extend these simulation algorithms to universal circuits by using stabilizer rank methods. We follow a previously known gadgetization procedure [9] to show that given a depth-d Clifford+T circuit with tT gates and whose two-qubit gates act along edges of a planar graph, we can sample from its output distribution in time widetildeO(20.7926tn5/2t6d3) and can compute output probabilities in time widetildeO(20.3963tn3/2t6d3). Previous work [9,6], applied to the case d=O(logn), gives algorithms for sampling in time O(20.3963tn6t6) and computation of output probabilities in time O(20.3963tn3t3). Our sampling algorithm offers improved scaling in n but poorer scaling in the exponential term, while our algorithm for computing output probabilities offers improved scaling in n with identical scaling in the exponential term.