Linear growth of quantum circuit complexity

Abstract

The complexity of quantum states has become a key quantity of interest across various subfields of physics, from quantum computing to the theory of black holes. The evolution of generic quantum systems can be modelled by considering a collection of qubits subjected to sequences of random unitary gates. Here we investigate how the complexity of these random quantum circuits increases by considering how to construct a unitary operation from Haar-random two-qubit quantum gates. Implementing the unitary operation exactly requires a minimal number of gates—this is the operation’s exact circuit complexity. We prove a conjecture that this complexity grows linearly, before saturating when the number of applied gates reaches a threshold that grows exponentially with the number of qubits. Our proof overcomes difficulties in establishing lower bounds for the exact circuit complexity by combining differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits.

The volume of the black hole interior at late times

“…. 16 This constant piece actually diverges as 1/∆. In the computation where we take ∆ → 0 first, this divergence shows up as [δ(ω)] 2 , where one factor of δ(ω) comes from (4.2), while the other comes from a contact term in ρ(s1)ρ(s2) , see (4.4).…”

“…1 The most important rationale behind this conjecture is that the black hole interior grows for a very long time, past when conventional probes, like correlation functions or entanglement entropy, have already reached their thermal equilibrium value. Complexity, on the other hand, is expected to grow in chaotic systems at the same rate as the volume of the interior, until times exponentially large in the entropy, after which it saturates in a complexity plateau [11][12][13][14][15][16]. Up to this point, it has been an open problem to establish whether the volume of the black hole interior exhibits a similar saturation at late times.…”

Quantum Computational Complexity -- From Quantum Information to Black Holes and Back

“…Of course, in order to reach this conclusion it is not enough to argue that the growth is generically linear, but one has to prove it. This has been done recently in [145], for the case of random circuits built from two-qubit gates, where each gate is drawn randomly according to the Haar measure on SU (4). The proof is basically a refinement of the counting argument, and it shows that the complexity is bounded below by a linear function of time, with probability 1.…”

Short Proofs of Linear Growth of Quantum Circuit Complexity

“…Recently, Ref. [4] proved a version of the above conjecture. Since all n-qubit gates can be synthesized by 2-qubit gates drawn from SU( 22 ), we define the complexity C(g) of a n-qubit gate g ∈ SU(2 n ) as the number of 2-qubit gates in a minimal synthesis.…”

“…U B (k), as a semialgebraic set, can be decomposed as disjoint union of smooth manifolds, and has a well-behaved dimension theory [4,5]. Denote d B (k) to be its dimension.…”