Linear growth of quantum circuit complexity

Abstract

Quantifying quantum states' complexity is a key problem in various subfields of science, from quantum computing to black-hole physics. We prove a prominent conjecture by Brown and Susskind about how random quantum circuits' complexity increases. Consider constructing a unitary from Haar-random two-qubit quantum gates. Implementing the unitary exactly requires a circuit of some minimal number of gates - the unitary's exact circuit complexity. We prove that this complexity grows linearly with the number of random gates, with unit probability, until saturating after exponentially many random gates. Our proof is surprisingly short, given the established difficulty of lower-bounding the exact circuit complexity. Our strategy combines differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits.

Figures (7)

• Figure 1: a. The complexity has been conjectured to grow linearly under random quantum circuits until times exponential in the number $n$ of qubits PhysRevD.97.086015. b. The blue region depicts part of the space of $n$-qubit unitaries. A unitary $U$ has a complexity that we define as the minimal number of two-qubit gates necessary to effect $U$ (green jagged path; each path segment represents a gate). Nielsen's complexity nielsen2005geometricnielsen2006quantumNielsen_06_Optimaldowling2008geometry, involved in Ref. PhysRevD.97.086015, attributes a high metric cost to directions associated with nonlocal operators. In this geometry, the unitary's complexity is the shortest path that connects $\mathbb{1}$ to $U$ (red line). Nielsen's geometry suggests the toolbox of differential geometry, avoiding circuits' discreteness. The circuit complexity upper-bounds Nielsen's complexity; opposite bounds hold for approximate circuit complexity dowling2008geometry.
• Figure 2: Our result relies on architectures and their backwards light cones. (a) An architecture specifies how $R$ 2-qubit gates are arranged in an $n$-qubit circuit. The gates need not be applied to neighboring qubits, though they are depicted this way for convenience. Our result involves blocks with the following property: The block contains a qubit reachable from each other qubit via a path (red dashed line), possibly unique to the latter qubit, that passes only through gates in the block. (b) The brickwork architecture interlaces layers of gates on a one-dimensional (1D) chain. In a 1D architecture with geometrically local gates, such as the brickwork architecture, each block has a backwards light cone (light-red region) that touches the qubit chain's edges. In the brickwork architecture, a minimal backwards-light-cone--containing block consists of ${\sim} n^2$ gates. (c) The staircase architecture, too, acts on a 1D qubit chain. The circuit consists of layers in which $n-1$ gates act on consecutive qubit pairs. A minimal backwards-light-cone--containing block consists of$n-1$ gates.
• Figure 3: The $R$-gate architecture $A$ is associated with a contraction map $F^A$. $F^A$ maps a list of input gates (a point in $[\mathrm{SU}(4)]^{\times R}$) to an $n$-qubit unitary $U$ in $\mathrm{SU}(2^n)$. The unitary results from substituting the gates into the architecture. $F^A$ has an image $\mathcal{U}(A)$, which consists of the unitaries implementable with the architecture. $A$ has an accessible dimension, $d_A$, equal to the dimension of $\mathcal{U}(A)$. Our core technical result is that $d_A$ grows linearly with $R$. To bridge this result to complexity, consider an arbitrary architecture $A'$ formed from fewer gates than a constant fraction of $R$. Such an architecture's accessible dimension satisfies $d_{A'} < d_A$, as we show. Therefore, every unitary in $\mathcal{U}(A)$ has a complexity linear in $R$, except for a measure-0 set. The proof relies on algebraic geometry. A key concept is the rank of $F^A$ at a point. The rank counts the local degrees of freedom in the image (orange arrows).
• Figure 4: Our core technical result is a lower bound on the accessible dimension (see Fig. \ref{['fig_Contraction_map']}). We prove this bound using a construction based on Clifford circuits. (a) Each gate $U_j$ is perturbed with a unitary $e^{i \epsilon_{j,k} S_k}$, generated by a 2-qubit Pauli operator $S_k$ and parameterized with an infinitesimal $\epsilon_{j,k} \in \mathbb{R}$. Perturbing the gate perturbs the $n$-qubit unitary, turning $U$ into $\tilde{U} \approx U$. (b) A key quantity is the derivative of $\tilde{U}$ with respect to a parameter $\epsilon_{j,k}$, evaluated at $U$. Taking this derivative is equivalent to inserting the Pauli string $S_k$ immediately after the gate $U_j$. (c) The derivative depicted in panel (b) is equivalent to following the circuit with a Hermitian operator $K_{j,k}$ [Eq. \ref{['eq_Decomp_Perturb']}]. The operator $K_{j,k}$ results from conjugating $S_k$ with the gates after $U_j$. If the circuit consists of Clifford gates, then $K_{j,k}$ is a Pauli string, since Clifford gates map the Pauli strings to Pauli strings. Therefore, a perturbation of $U_j$ in the direction of $S_k$ results in a perturbation of the resulting unitary $U$ in the direction of $K_{j,k}$ in $\mathrm{SU}(2^n)$. (d) The following is true of every backwards-light-cone--containing block and every Pauli string $P$ (leftmost green squares): The block's gates can be chosen to be Cliffords that map $P$ to a single-site $Z$. The Clifford gates first map $P$ to a Pauli string that acts nontrivially on fewer qubits (pale green squares), then to a Pauli string on fewer qubits, and so on until the Pauli string dwindles to one qubit (rightmost green square). (e) Our lower bound is proven by recursion. Consider an architecture $A_{T'}$, formed from $T' < 4^n - 1$backwards-light-cone--containing blocks, whose accessible dimension is $\geq T'$. There exist gates $U_1, U_2, \ldots,U_{R'}$ such that that $T'$ linearly independent Pauli operators $K'_{j_m,k_m}$ (wherein $m = 1, 2, \ldots, T'$) result from perturbing the gates, as described in (a)--(c). Consider a Pauli operator $P$ that is not in $\{ K'_{j_m,k_m} \}$. We can append to $A_{T'}$ a backwards-light-cone--containing block, formed from Clifford gates, that maps $P$ to a single-site $Z$, as depicted in panel (d). This $Z$ is an important direction in $\mathrm{SU}(2^n)$: Consider perturbing the block's final gate via the procedure in (a)--(c). The image $\mathcal{U}(A_{T'})$ is perturbed, as a result, in the direction $Z$. Thus, $T'+1$ linearly independent Pauli operators (the operators $K'_{j_m,k_m}$ and $P$) result from perturbing gates in the extended circuit. Therefore, the extended circuit's accessible dimension is $\geq T'+1$.
• Figure 5: Three-part map used in the proof of Lemma \ref{['lemma:locus']}. $H_j$ denotes the $j^{\rm th}$ two-qubit Hermitian operator, $U_j$ denotes the $j^{\rm th}$ two-qubit unitary, and $H$ denotes an $n$-qubit Hermitian operator.

Theorems & Definitions (30)

• Definition 1: Architecture
• Definition 2: Random quantum circuit
• Definition 3: Exact circuit complexities
• Theorem 1: Linear growth of complexity
• Proposition 1: Lower bound on accessible dimension
• Lemma 1: Low-rank locus
• Definition 4: Algebraic set
• Definition 5: Zariski topology
• Definition 6: Irreducible set
• Definition 7: Dimension of algebraic sets