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A geometric approach to quantum circuit lower bounds

Michael A. Nielsen

TL;DR

The paper develops a geometric framework that connects exact quantum circuit size to geodesic length under right-invariant Finsler metrics on $SU(2^n)$. By formulating a Hamiltonian-control problem and constructing several metrics ($F_1,F_2,F_p,F_q$), it derives a geodesic equation and shows how minimal lengths bound circuit size via $d_F(I,U)\le m_{\cal G}(U)$; it also introduces Pauli geodesics and links the minimal Pauli geodesic for diagonal unitaries to the closest vector problem (CVP), proving exponential-length behavior for most such unitaries. The work provides a concrete, though still partial, path toward proving quantum circuit lower bounds and offers a rich set of tools (geodesic computation, coordinate changes, and symmetry-based Pauli geodesics) with important caveats and directions for future research. Overall, it reframes circuit complexity questions in a geometric light, suggesting new avenues for both lower-bound proofs and potential classical simulations under certain equivalence assumptions.

Abstract

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on SU(2^n). The geodesic curves of such a metric have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size, and give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call Pauli geodesics, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

A geometric approach to quantum circuit lower bounds

TL;DR

The paper develops a geometric framework that connects exact quantum circuit size to geodesic length under right-invariant Finsler metrics on . By formulating a Hamiltonian-control problem and constructing several metrics (), it derives a geodesic equation and shows how minimal lengths bound circuit size via ; it also introduces Pauli geodesics and links the minimal Pauli geodesic for diagonal unitaries to the closest vector problem (CVP), proving exponential-length behavior for most such unitaries. The work provides a concrete, though still partial, path toward proving quantum circuit lower bounds and offers a rich set of tools (geodesic computation, coordinate changes, and symmetry-based Pauli geodesics) with important caveats and directions for future research. Overall, it reframes circuit complexity questions in a geometric light, suggesting new avenues for both lower-bound proofs and potential classical simulations under certain equivalence assumptions.

Abstract

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on SU(2^n). The geodesic curves of such a metric have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size, and give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call Pauli geodesics, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

Paper Structure

This paper contains 38 sections, 13 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Motivating ideas
  4. Structure and main results
  5. Prior work
  6. The Hamiltonian control problem and metrics on manifolds
  7. The Hamiltonian control problem
  8. Cost functions and right-invariant local metrics on manifolds
  9. Desired properties of the cost function $f$
  10. Geometric reformulation of the cost function
  11. Minimal curves and lower bounds
  12. Geodesics and Finsler geometry
  13. Examples of local metrics whose minimal length curves provide lower bounds on circuit size
  14. Examples of local metrics whose minimal length curves provide lower bounds on circuit size
  15. Properties of $F_2$ and $F_q$
  16. ...and 23 more sections

Key Result

Proposition 1

Let $U(t)$ be a smooth curve in $SU(2^n)$. Then (a) $i \frac{dU}{dt} U^\dagger$ is Hermitian, and (b) the natural $U(t)$-adapted co-ordinates $y^\sigma$ for the tangent to the curve, $[U]_t \in T_{U(t)} SU(2^n)$, are determined by the equationNote the convention, used here and throughout, that the $ In particular, we have $y^\sigma = i \hbox{tr}(\sigma \, dU/dt \, U^\dagger) / 2^n$.

Theorems & Definitions (13)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3: BCH formula
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • ...and 3 more