Ansatz Expressibility

An ansatz $U(\boldsymbol{\theta})$ defines a family of states $\{|\psi(\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta})|0\rangle\}$ as the parameter vector $\boldsymbol{\theta}$ ranges over its domain. Expressibility asks a sharp question: how large, and how uniformly, does this family cover the Hilbert space? If the true ground state of your Hamiltonian lies outside the reachable set, no optimizer — however clever — can ever find it. Expressibility is therefore a necessary (though not sufficient) condition for a successful variational algorithm.

The reachable set

For an $n$ -qubit system the pure states live on a $2^{n}-1$ complex-dimensional projective space. A parameterized circuit with $p$ real parameters traces out a manifold of dimension at most $p$ inside that space. Two consequences follow immediately:

If $p$ is smaller than the dimension of the space, the ansatz is underparameterized and generically misses most states.
Even when $p$ is large, the manifold can fold back on itself, leaving regions unreachable — redundant parameters do not buy coverage.

A useful sanity check is to count. The unitary group $\mathrm{SU}(2^n)$ has $4^n - 1$ real parameters. A hardware-efficient ansatz of $L$ layers on $n$ qubits with three rotation angles per qubit per layer has roughly $3nL$ parameters. To even have a chance of universality you need $3nL \gtrsim 4^n - 1$ , which grows exponentially in $n$ .

Measuring expressibility with state overlaps

Sim, Johnson, and Aspuru-Guzik proposed a concrete, samplable metric. Draw two parameter vectors $\boldsymbol{\theta}, \boldsymbol{\phi}$ at random and record the fidelity $F = |\langle\psi(\boldsymbol{\theta})|\psi(\boldsymbol{\phi})\rangle|^2$ . Repeating this builds an empirical distribution $P_{\text{ansatz}}(F)$ . Compare it to the distribution you would get if the states were drawn uniformly (Haar-randomly) from the Hilbert space, which for dimension $d = 2^n$ is

P_{\text{Haar}}(F) = (d-1)\,(1-F)^{\,d-2}.

Expressibility is then quantified as the Kullback–Leibler divergence

\text{Expr} = D_{\mathrm{KL}}\!\bigl(P_{\text{ansatz}}(F)\,\|\,P_{\text{Haar}}(F)\bigr).

A smaller divergence means the ansatz scatters states almost as uniformly as a Haar-random unitary — high expressibility. A large divergence means the circuit clusters its outputs in a small corner of Hilbert space — low expressibility.

Worked single-qubit example

Consider the one-parameter ansatz $|\psi(\theta)\rangle = R_Y(\theta)|0\rangle = \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle$ . With only a real amplitude it traces a great circle on the Bloch sphere — the $xz$ -plane — and can never reach states with a complex relative phase such as $|{+i}\rangle$ . It is expressively limited.

Add a second rotation, $|\psi(\theta,\phi)\rangle = R_Z(\phi)R_Y(\theta)|0\rangle$ . Now $\theta$ sets the polar angle and $\phi$ the azimuth, so the two parameters sweep the entire Bloch sphere: this ansatz is universal for one qubit. Counting matches intuition — the single-qubit pure state space is 2-real-dimensional, and we used exactly two parameters.

Why more is not always better

Expressibility trades against two other concerns you will study next:

Trainability. Highly expressible ansätze that approximate Haar-random unitaries suffer from barren plateaus — exponentially flat cost landscapes. The very property that lets a circuit reach everywhere makes its gradients vanish.
Hardware cost. Each extra layer adds noisy two-qubit gates. On NISQ devices a deep, maximally expressible circuit may be unusable in practice.

The art of ansatz design is choosing a circuit expressible enough to contain the answer, but structured enough — by symmetry, locality, or problem knowledge — to stay trainable and shallow.

Key takeaways

Expressibility is how well, and how uniformly, an ansatz covers Hilbert space.
It is bounded by the parameter count $p$ and degraded by redundancy.
A practical metric is the KL divergence of the state-overlap distribution from the Haar distribution $P_{\text{Haar}}(F) = (d-1)(1-F)^{d-2}$ .
More expressibility is not free: it competes with trainability and circuit depth.