Variational Quantum Dynamics and Imaginary-Time Evolution

Simulating a quantum system means tracking a state as it evolves — under $\mathrm{e}^{-\mathrm{i}Ht}$ in real time, or the Wick-rotated $\mathrm{e}^{-H\tau}$ in imaginary time — and both flows live in a Hilbert space whose dimension doubles with every qubit, so writing the state exactly is hopeless past a few dozen qubits. Variational quantum dynamics makes the problem finite by one act of discipline: it forbids the state to leave a low-dimensional surface, the manifold $\mathcal{M}=\{|\psi(\boldsymbol\theta)\rangle:\boldsymbol\theta\in\mathbb{R}^p\}$ swept out by a parameterized circuit as its $p$ angles vary. The wavefunction is no longer $2^n$ amplitudes but $p$ classical numbers, and the dynamics becomes an ODE for how those numbers move. The rule for how they move is a variational principle, and it is pure geometry. The true evolution wants to push the state in some direction — the exact velocity vector — but most of that direction points off the manifold, where the ansatz cannot follow. So you project: keep the part of the exact velocity lying in the manifold's tangent space, discard the rest. The projection is a least-squares fit carried out with the manifold's own ruler, the Fubini–Study metric, and its output is one compact linear system solved every timestep, $A(\boldsymbol\theta)\,\dot{\boldsymbol\theta}=C(\boldsymbol\theta)$ , in which $A$ is the geometry and $C$ is the force. For imaginary time the force is minus half the energy gradient, so the flow is steepest descent on the energy — this is how a quantum computer prepares a ground state, and why the method is one of the most-studied primitives on near-term hardware.

Key takeaways

Variational quantum dynamics confines the wavefunction to a $p$ -dimensional manifold $\{|\psi(\boldsymbol\theta)\rangle\}$ set by a circuit's angles, turning real-time (Schrödinger) or imaginary-time (Wick-rotated) evolution into a classical equation of motion $A(\boldsymbol\theta)\,\dot{\boldsymbol\theta}=C(\boldsymbol\theta)$ .
The geometry is the quantum geometric tensor $G_{ij}$ ; its real part is the Fubini–Study metric $A$ (the matrix in the EOM), its imaginary part the Berry curvature.
Imaginary time makes the force $C_i=-\tfrac12\partial_i\langle H\rangle$ , so the flow is $\dot{\boldsymbol\theta}=-\tfrac12 A^{-1}\nabla\langle H\rangle$ — exactly quantum natural-gradient descent, the way variational imaginary-time evolution (VITE) prepares ground states; $A_{ij}$ and $C_i$ are estimated on hardware (Hadamard-test and parameter-shift circuits) and the small system solved classically each step.
$A$ is generically singular and ill-conditioned, so it must be regularized ( $A+\epsilon\mathbb{1}$ or pseudo-inverse) — naive inversion amplifies shot noise catastrophically.
Even with perfect circuits, accuracy is capped by the McLachlan residual $\|R\|$ : the exact velocity that leaks out of the tangent space — the dynamical face of limited expressibility.
Key idea: replace exact evolution in exponentially large Hilbert space by a flow on a finite-dimensional parameter manifold — project the exact velocity onto the tangent space with the Fubini–Study metric to get $A\dot{\boldsymbol\theta}=C$ , where $A$ is the geometry and $C$ is the force (the energy gradient for imaginary time, $-\mathrm{i}$ times it for real time).

The core insight

Replace exact evolution in exponentially large Hilbert space by a flow on a finite-dimensional parameter manifold: project the exact velocity vector onto the tangent space using the Fubini–Study metric, giving $A(\boldsymbol\theta)\dot{\boldsymbol\theta}=C(\boldsymbol\theta)$ , where $A$ is the geometry and $C$ is the force (energy gradient for imaginary time, $-\mathrm{i}$ times it for real time).

Where this lives today

Variational imaginary-time evolution is one of the most actively benchmarked primitives in near-term quantum computing. McArdle and collaborators demonstrated it for small molecules such as H₂ and LiH, and the related quantum imaginary-time evolution (QITE) of Motta and co-workers — which sidesteps the metric inversion by fitting a local unitary to the McLachlan-equivalent update — has run on trapped-ion and superconducting devices to prepare spin-chain ground states and sample thermal (Gibbs) states $\propto\mathrm{e}^{-\beta H}$ . The quantum natural gradient VITE reveals is now a standard optimizer in PennyLane and Qiskit, the metric cached across steps to amortize its $O(p^2)$ cost. Real-time variational dynamics extracts spectral functions by Fourier-transforming dynamical correlators. The open problems connect to the rest of the chapter: the metric costs $O(p^2)$ expectation values per step; the condition number of $A$ worsens as circuits deepen; and barren plateaus flatten the force and collapse the metric together, so VITE inherits VQE's trainability wall.

The physical picture

Picture the exact quantum state as a point on the unit sphere of a colossal Hilbert space, moving with a velocity Schrödinger's equation dictates. Your ansatz is a thin curved sheet — the manifold — stitched into that sphere by the circuit, and you are pinned to it. The best you can do is move along the sheet in whatever direction most nearly matches the true velocity, where "most nearly" is measured by the Fubini–Study metric, which reports how distinguishable two nearby states on the sheet are. The variational principle is just least-squares fitting of the on-sheet velocity to the off-sheet target, and its output is the linear system $A\dot{\boldsymbol\theta}=C$ . The first figure is that projection made literal.

Figure

The variational principle is least-squares fitting: project the exact imaginary-time velocity −(H−⟨H⟩)ψ onto the tangent space of the ansatz manifold to obtain the equation of motion A θ̇ = C, and the leftover residual ‖R‖ that cannot be represented is the irreducible expressibility error

Imaginary time turns this geometric fitting into a ground-state algorithm. Real evolution $\mathrm{e}^{-\mathrm{i}Ht}$ only spins each eigencomponent by a phase $\mathrm{e}^{-\mathrm{i}E_n t}$ ; the Wick rotation $t\to-\mathrm{i}\tau$ converts those phases into real exponentials $\mathrm{e}^{-E_n\tau}$ , so high-energy components decay and, after renormalizing, the state slides downhill toward the ground state. On the manifold that slide is steepest descent — but measured with the metric ruler $A$ , not the arbitrary Euclidean ruler of the raw angles, which is why it can outrun a plain optimizer. The second figure shows the descent in the simplest, fully checkable case.

Figure

Variational imaginary-time evolution of one qubit with H = ½Z started from the equal-superposition state: the energy ⟨H⟩(τ) = −½ tanh τ slides monotonically down the metric-steepest path to the exact ground-state energy E₀ = −½ as τ → ∞, reaching within 0.4 percent by τ = 3

The equation $A\dot{\boldsymbol\theta}=C$ is the quantum sibling of natural-gradient learning, with $A$ the Fisher information and $A^{-1}C$ the natural gradient. Its one honest weakness is the amber residual in the first figure: the scheme works only insofar as the true velocity nearly lies in the tangent space, and when the state grows entanglement the ansatz cannot represent, that residual is leakage the method can never recover — the dynamical signature of limited expressibility.

The core equations

The starting objects are the state the circuit prepares and the tangent vectors from differentiating it in each angle; these span the tangent space onto which everything is projected.

|\psi(\boldsymbol\theta)\rangle = U_p(\theta_p)\cdots U_1(\theta_1)\,|\psi_0\rangle,\qquad |\partial_i\psi\rangle \equiv \frac{\partial}{\partial\theta_i}|\psi(\boldsymbol\theta)\rangle \qquad (1)

where: $U_k(\theta_k)$ — parameterized gate, typically $\mathrm{e}^{-\mathrm{i}\theta_k G_k/2}$ with Hermitian generator $G_k$ ; $|\psi_0\rangle$ — fixed reference, e.g. $|0\rangle^{\otimes n}$ ; $|\partial_i\psi\rangle$ — tangent vector; $\boldsymbol\theta$ — the $p$ real parameters. Reading: the circuit is a chart, $\boldsymbol\theta$ its coordinates, the $|\partial_i\psi\rangle$ its tangent basis. The tangent-vector overlaps, with the global-phase direction projected out, define the geometry — one complex Hermitian tensor packaging metric and curvature.

G_{ij} = \langle\partial_i\psi|\partial_j\psi\rangle - \langle\partial_i\psi|\psi\rangle\langle\psi|\partial_j\psi\rangle \qquad (2)

where: $G_{ij}$ — quantum geometric tensor; $A_{ij}=\mathrm{Re}\,G_{ij}$ — Fubini–Study metric (real, symmetric, PSD), a quarter of the quantum Fisher information and the matrix in the EOM; $B_{ij}=\mathrm{Im}\,G_{ij}$ — Berry curvature (antisymmetric). Reading: the subtraction removes the unphysical global-phase direction, since physical states are rays. Imposing McLachlan's principle in real time projects $-\mathrm{i}H|\psi\rangle$ onto the tangent space; the imaginary part of the Hamiltonian element becomes the force.

\sum_j \mathrm{Re}\,G_{ij}\,\dot\theta_j = -\,\mathrm{Im}\,\langle\partial_i\psi|H|\psi\rangle \quad\Longleftrightarrow\quad A\,\dot{\boldsymbol\theta}=C^{\mathrm{(R)}} \qquad (3)

where: $A_{ij}=\mathrm{Re}\,G_{ij}$ — Fubini–Study metric; $C^{\mathrm{(R)}}_i=-\mathrm{Im}\,\langle\partial_i\psi|H|\psi\rangle$ — real-time force; $H$ — Hamiltonian; $\dot{\boldsymbol\theta}$ — velocity solved each step. Reading: real-time flow conserves energy and rotates the state, so the force is the imaginary part of the Hamiltonian element. Wick-rotating to imaginary time, the exact velocity becomes $-(H-\langle H\rangle)|\psi\rangle$ and the projection keeps its real part — the equation VITE integrates.

\sum_j \mathrm{Re}\,G_{ij}\,\dot\theta_j = -\,\mathrm{Re}\,\langle\partial_i\psi|H|\psi\rangle \quad\Longleftrightarrow\quad A\,\dot{\boldsymbol\theta}=C^{\mathrm{(I)}} \qquad (4)

where: $C^{\mathrm{(I)}}_i=-\mathrm{Re}\,\langle\partial_i\psi|H|\psi\rangle=-\tfrac12\,\partial_i\langle H\rangle$ — imaginary-time force, exactly minus half the energy gradient; $\tau$ — imaginary time, $t=-\mathrm{i}\tau$ . Reading: the force is the energy gradient, so imaginary-time flow is energy minimization — the bridge to ground states. Solving (4) with $C^{\mathrm{(I)}}=-\tfrac12\nabla\langle H\rangle$ makes the natural-gradient structure explicit: the metric preconditions the ordinary energy gradient.

\dot{\boldsymbol\theta} = -\tfrac12\,A^{-1}\,\nabla_{\boldsymbol\theta}\langle H\rangle \qquad (5)

where: $A^{-1}$ — inverse metric, the natural-gradient preconditioner; $\nabla_{\boldsymbol\theta}\langle H\rangle$ — Euclidean energy gradient (parameter-shift estimable). Reading: discretized with step $\eta$ , this is $\boldsymbol\theta\leftarrow\boldsymbol\theta-\eta\,A^{-1}\nabla\langle H\rangle$ . All of this descends from one scalar cost, the squared residual $\mathcal{L}_{\mathrm{McL}}=\|(\partial_\tau+(H-\langle H\rangle))|\psi\rangle\|^2$ minimized over the velocities (built explicitly in the first derivation). In practice $A$ is rank-deficient with smallest eigenvalues at the shot-noise level, so it is never inverted bare; a Tikhonov shift stabilizes the solve at the price of a controlled bias.

A^{\mathrm{reg}} = A + \epsilon\,\mathbb{1},\qquad \dot{\boldsymbol\theta} = (A+\epsilon\mathbb{1})^{-1}\,C \qquad (6)

where: $\epsilon$ — regularization strength trading bias against noise; $\mathbb{1}$ — $p\times p$ identity. Reading: as $\epsilon\to0$ one recovers the ideal EOM but with diverging noise gain; the pseudo-inverse $A^{+}$ is the $\epsilon\to0^{+}$ limit on the well-conditioned subspace.

Derivation AThe McLachlan principle as a least-squares projection

The goal is to show that minimizing the McLachlan residual norm over the parameter velocities is not an arbitrary recipe but the unique least-squares projection of the exact imaginary-time velocity onto the tangent space, delivering the linear system $A\dot{\boldsymbol\theta}=C$ with the Fubini–Study metric as its matrix and the energy element as its force. We build the residual, square it, differentiate over the velocities, and read off geometry and force from the surviving terms.

Step 1. Write the residual between the velocity the manifold can supply and the one exact imaginary-time dynamics demands: $|R\rangle = \sum_j \dot\theta_j|\partial_j\psi\rangle + (H-\langle H\rangle)|\psi\rangle$ . Why: the manifold moves only within $\mathrm{span}\{|\partial_j\psi\rangle\}$ , while the exact target is $-(H-\langle H\rangle)|\psi\rangle$ ; the residual is the mismatch to minimize.

Step 2. Form the scalar cost $\mathcal{L}_{\mathrm{McL}}=\langle R|R\rangle$ , a real quadratic in the $p$ velocities. Why: squaring the norm turns "track the true velocity as closely as possible" into a single differentiable objective — the whole content of the variational principle.

Step 3. Impose $\partial\mathcal{L}_{\mathrm{McL}}/\partial\dot\theta_i=0$ ; differentiating the real quadratic form gives $\sum_j \mathrm{Re}\,\langle\partial_i\psi|\partial_j\psi\rangle\,\dot\theta_j + \mathrm{Re}\,\langle\partial_i\psi|(H-\langle H\rangle)|\psi\rangle = 0$ . Why: this is the least-squares normal equation — the residual is made orthogonal, in the real inner product, to every tangent direction.

Step 4. With the global-phase terms folded in, the first sum is $\sum_j\mathrm{Re}\,G_{ij}\dot\theta_j = A\dot{\boldsymbol\theta}$ , and the $\langle H\rangle$ piece cancels against the phase-projection terms of $G_{ij}$ , leaving $A\dot{\boldsymbol\theta} = -\mathrm{Re}\,\langle\partial_i\psi|H|\psi\rangle = C$ . Why: the gauge structure removes the normalization piece, so the EOM is coordinate-free. *Verified: when the tangent space contains $H|\psi\rangle$ , choosing $\dot{\boldsymbol\theta}=A^{-1}C$ drives $\langle R|R\rangle=0$ , and the variational trajectory reproduces exact imaginary-time evolution — confirmed by direct substitution in the single-qubit case below.*

Go deeper · Derivation B · 1 minWhy imaginary time slides downhill in energy

Derivation BWhy imaginary time slides downhill in energy

The goal is to prove that the variational imaginary-time trajectory is a genuine descent flow: the energy $\langle H\rangle$ never increases along it, and the flow halts precisely at a variational stationary point — the same fixed point a converged VQE run reaches. This licenses VITE as a ground-state algorithm rather than a heuristic, and the proof rests only on the positive-semidefiniteness of $A$ , so it holds for any ansatz.

Step 1. Differentiate the energy by the chain rule: $\dfrac{\mathrm{d}\langle H\rangle}{\mathrm{d}\tau} = \sum_i \partial_i\langle H\rangle\,\dot\theta_i = \nabla\langle H\rangle^{\!\top}\dot{\boldsymbol\theta}$ . Why: the state moves only through its parameters, so all of the energy change is carried by the velocity.

Step 2. Insert the EOM in natural-gradient form, $\dot{\boldsymbol\theta}=-\tfrac12 A^{-1}\nabla\langle H\rangle$ . Why: the imaginary-time force is minus half the energy gradient, preconditioned by the inverse metric.

Step 3. Collect the quadratic form: $\dfrac{\mathrm{d}\langle H\rangle}{\mathrm{d}\tau} = -\tfrac12\,\nabla\langle H\rangle^{\!\top} A^{-1}\nabla\langle H\rangle \le 0$ . Why: $A\succeq0$ , so $A^{-1}$ (on its support) is too, making the form non-negative and the energy derivative non-positive.

Step 4. The flow stops when $\nabla\langle H\rangle$ lies in $\ker A$ or vanishes — a stationary point of the variational energy. Why: this is exactly what a VQE optimizer targets, so VITE and VQE share fixed points but reach them along different paths. *Verified: equality $\mathrm{d}\langle H\rangle/\mathrm{d}\tau=0$ holds only at stationarity; away from it the descent is strict, guaranteeing progress toward the best ground-state approximation on the manifold.*

Go deeper · Derivation C · 1 minOne qubit solved in closed form

Derivation COne qubit solved in closed form

The goal is a fully checkable instance in which VITE reproduces exact imaginary-time evolution with no residual, so every abstract quantity — the metric $A$ , the force $C$ , the flow $\dot\theta$ , the long-time limit — is closed form and matches the exact operator $\mathrm{e}^{-H\tau}$ . We take $H=\tfrac12 Z$ with the one-parameter ansatz $|\psi(\theta)\rangle=\cos\tfrac{\theta}{2}|0\rangle+\sin\tfrac{\theta}{2}|1\rangle$ , a Bloch great circle containing both eigenstates, which guarantees exactness.

Step 1. The tangent vector is $|\partial_\theta\psi\rangle=\tfrac12(-\sin\tfrac\theta2|0\rangle+\cos\tfrac\theta2|1\rangle)$ , and since $\langle\psi|\partial_\theta\psi\rangle=0$ the phase term drops, giving $A=\tfrac14$ . Why: on a great circle the state and its derivative are orthogonal, so the metric is just the squared speed of the tangent vector.

Step 2. The energy is $\langle H\rangle=\tfrac12\cos\theta$ , so $\partial_\theta\langle H\rangle=-\tfrac12\sin\theta$ and $C=-\tfrac12\partial_\theta\langle H\rangle=+\tfrac14\sin\theta$ . Why: the two sign flips leave $C$ positive — a point where careless bookkeeping goes wrong.

Step 3. Solve $A\dot\theta=C$ : $\tfrac14\dot\theta=\tfrac14\sin\theta\Rightarrow\dot\theta=\sin\theta$ , unstable at $\theta=0$ ( $|0\rangle$ , $E=+\tfrac12$ ) and stable at $\theta=\pi$ ( $|1\rangle$ , $E=-\tfrac12$ ). Why: the force points the flow toward larger $\theta$ , i.e. toward $|1\rangle$ , the ground state of $\tfrac12 Z$ .

Step 4. The ODE gives $\tan\tfrac{\theta(\tau)}{2}=\mathrm{e}^{\tau}\tan\tfrac{\theta_0}{2}$ ; from $|+\rangle$ ( $\theta_0=\tfrac\pi2$ ) this is the clean law $\langle H\rangle(\tau)=-\tfrac12\tanh\tau$ . Normalizing $\mathrm{e}^{-H\tau}|+\rangle=\mathrm{e}^{-\tau/2}|0\rangle+\mathrm{e}^{+\tau/2}|1\rangle$ gives $\tan\tfrac\theta2=\mathrm{e}^{\tau}$ — identical. Why: the manifold contains $H|\psi\rangle$ , so the residual is zero and the trajectories coincide term by term. *Verified: $A=\tfrac14$ , $\dot\theta=\sin\theta$ , $\langle H\rangle(\tau)=-\tfrac12\tanh\tau\to-\tfrac12$ as $\tau\to\infty$ , matching normalized $\mathrm{e}^{-H\tau}|+\rangle$ exactly; this is the curve plotted in the second figure.*

Limiting cases

Variational dynamics interpolates between exact simulation and a frozen state, depending on how large and well-conditioned the manifold is.

Case	Result	Meaning
Full-rank, expressive manifold	$\mathrm{span}\{\vert\partial_i\psi\rangle\}\supseteq H\vert\psi\rangle$	The tangent space contains the exact velocity, so the projection is exact, $\\|R\\|=0$ , and the trajectory reproduces true evolution — the single-qubit case above.
Rank-deficient metric	$\det A=0\Rightarrow A^{+}\ (\text{Moore–Penrose})$	Redundant parameters make $A$ singular; the pseudo-inverse or $A+\epsilon\mathbb{1}$ projects onto the well-conditioned subspace at the cost of a controlled bias.
Imaginary-time long-time limit	$\tau\to\infty:\ \vert\psi(\boldsymbol\theta(\tau))\rangle\to\vert\psi_{\mathrm{var}}\rangle$	VITE converges to the best ground-state approximation on the manifold — the same fixed point as a successful VQE run, along the metric-steepest path.
Barren-plateau regime	$\mathrm{Var}[C_i]\sim 2^{-n},\ \lambda_{\min}(A)\to 0$	Exponentially flat landscapes make force and metric exponentially small, so shot noise swamps the system and the dynamics stall just as optimization does.

Common trapTreating the metric A as if it were always invertible

The commonest way to break a VITE implementation is to solve $A\dot{\boldsymbol\theta}=C$ by naive inversion. The Fubini–Study metric is generically rank-deficient — any redundant or symmetry-locked parameter drives an eigenvalue of $A$ to zero, and even the nonzero eigenvalues can be tiny. Since $A$ and $C$ are estimated with finite shots, inverting a near-singular $A$ multiplies the shot noise by the condition number, and the velocities explode into pure noise. Regularization (Tikhonov $A+\epsilon\mathbb{1}$ , a truncated pseudo-inverse, or an $L$ -curve-chosen $\epsilon$ ) is mandatory, not cosmetic. A subtler trap is conflating the variational principles: McLachlan, Dirac–Frenkel, and TDVP coincide for real parameters, but for complex parameterizations they give different equations of motion because they treat the imaginary part of the geometric tensor differently.

Worked examples

1 · Ground state of a qubit by imaginary-time descent (fully worked)

Take $H=\tfrac12 Z$ and the ansatz $|\psi(\theta)\rangle=\cos\tfrac\theta2|0\rangle+\sin\tfrac\theta2|1\rangle$ from $|+\rangle$ ( $\theta_0=\tfrac\pi2$ ). Integrate the VITE equation of motion; tabulate the energy, one Euler step, and the long-time limit.

Step 1. From the closed-form derivation, $A=\tfrac14$ and $C=-\tfrac12\partial_\theta\langle H\rangle=\tfrac14\sin\theta$ , so the EOM is $\tfrac14\dot\theta=\tfrac14\sin\theta$ , i.e. $\dot\theta=\sin\theta$ . With $p=1$ there is nothing to regularize.

Step 2. One explicit Euler step with $\Delta\tau=0.1$ : at $\theta_0=\tfrac\pi2$ , $\dot\theta=1$ , so $\theta_1=1.6708$ and $\langle H\rangle=\tfrac12\cos(1.6708)=\mathbf{-0.0499}$ . The exact law $-\tfrac12\tanh(0.1)=-0.0499$ agrees to four decimals, as it must on an exact manifold.

Step 3. Tabulate the energy from $\langle H\rangle(\tau)=-\tfrac12\tanh\tau$ : at $\tau=0.5$ , $-0.231$ ; at $\tau=1$ , $\mathbf{-0.381}$ ; at $\tau=2$ , $-0.482$ ; at $\tau=3$ , $\mathbf{-0.498}$ — monotone, and within $0.4\%$ of the ground-state energy by $\tau=3$ .

Step 4. As $\tau\to\infty$ , $\langle H\rangle\to\mathbf{-\tfrac12}$ and $\theta\to\pi$ , so $|\psi\rangle\to|1\rangle$ , the exact ground state of $\tfrac12 Z$ . VITE prepares it exactly here because the manifold contains $H|\psi\rangle$ ; the residual is zero at every $\tau$ .

2 · A singular metric and why regularization saves it

Use one qubit with two redundant angles, $|\psi\rangle=R_x(\theta_1)R_x(\theta_2)|0\rangle$ , generator $X$ on both. Show that $A$ is singular, then repair the solve with the pseudo-inverse and with Tikhonov regularization.

Step 1. Both gates share generator $X$ and commute, so $R_x(\theta_1)R_x(\theta_2)=R_x(\theta_1+\theta_2)$ and the tangent vectors are equal: $|\partial_1\psi\rangle=|\partial_2\psi\rangle=-\tfrac{\mathrm i}{2}X|\psi\rangle$ . Each has squared norm $\tfrac14$ and $\langle\psi|X|\psi\rangle=0$ , giving $A=\tfrac14\begin{pmatrix}1&1\\1&1\end{pmatrix}$ . Its eigenvalues are $\tfrac12$ (along $(1,1)$ ) and $0$ (along $(1,-1)$ ), so $\det A=0$ . The zero mode changes $\theta_1-\theta_2$ without moving the state.

Step 2. With $\varphi=\theta_1+\theta_2$ , $\langle H\rangle=\tfrac12\cos\varphi$ , so $C=-\tfrac12\nabla\langle H\rangle=\tfrac14\sin\varphi\,(1,1)^{\top}$ — force and true motion live entirely in the well-conditioned $(1,1)$ eigenspace.

Step 3. The pseudo-inverse is $A^{+}=\begin{pmatrix}1&1\\1&1\end{pmatrix}$ , giving $\dot{\boldsymbol\theta}=A^{+}C=\tfrac12\sin\varphi\,(1,1)^{\top}$ , so $\dot\varphi=\sin\varphi$ — exactly the correct one-parameter flow from Example 1. The pseudo-inverse ignores the zero mode.

Step 4. With Tikhonov instead, $A+\epsilon\mathbb{1}$ has eigenvalues $\tfrac12+\epsilon$ and $\epsilon$ , condition number $(\tfrac12+\epsilon)/\epsilon$ — 51 at $\epsilon=0.01$ , diverging as $\epsilon\to0$ . Since $C$ lies in the $\tfrac12+\epsilon$ eigenspace, $\dot{\boldsymbol\theta}=\tfrac{\tfrac14\sin\varphi}{\tfrac12+\epsilon}(1,1)^{\top}\to\tfrac12\sin\varphi\,(1,1)^{\top}$ as $\epsilon\to0$ , matching the pseudo-inverse with finite noise gain. The physics lives in one eigendirection; regularization inverts without amplifying noise in the empty one.

Physics of the portrait

The equation $A\dot{\boldsymbol\theta}=C$ carries three names because three generations found it from three directions: quantum mechanics needed a way to evolve approximate wavefunctions, and each era's tool — Dirac's action, Frenkel's overlap, McLachlan's residual — turned out to be the same projection wearing different clothes.

Origins · 1 mintime-dependent variational principle and how this was discovered

time-dependent variational principle

the projection δ⟨ψ · (i∂ₜ − H)

The idea that an approximate state should evolve by a variational rule is nearly as old as quantum mechanics. Paul Dirac introduced it in 1930, in his note on the Thomas atom, to ground the time-dependent Hartree–Fock equations; Yakov Frenkel, in his 1934 textbook, cast it in the form still quoted today, $\langle\delta\psi|(\mathrm{i}\partial_t-H)|\psi\rangle=0$ — the exact velocity projected against every allowed variation. For three decades it powered multiconfigurational methods in chemistry and nuclear physics, but it assumed a holomorphically parameterized state. In 1964 A. D. McLachlan removed that assumption: rather than stationarizing an action he minimized the residual norm $\|(\partial_t+\mathrm{i}H)|\psi\rangle\|$ over real parameter velocities, a least-squares form that works for any real parameterization and reduces to Dirac–Frenkel when the state is holomorphic — the version that survives on hardware. When Li and Benjamin (2017) and then McArdle, Yuan and collaborators (2019) asked how to run imaginary-time evolution on a NISQ device, McLachlan's residual gave the estimable system $A\dot{\boldsymbol\theta}=C$ — metric by Hadamard tests, force by parameter shifts — and Yuan and co-workers (2019) unified all three principles into one theory of variational quantum simulation, the same object Stokes and collaborators (2020) named the quantum natural gradient. What Dirac wrote for a helium atom is now how a quantum computer chases a ground state.

History

1930

— Dirac casts time-dependent Hartree–Fock as a variational principle

1934

— Frenkel writes it as δ⟨ψ|(i∂ₜ − H)|ψ⟩ = 0

1964

— McLachlan minimizes the residual norm, freeing it from holomorphic parameters

2011

— Haegeman applies the time-dependent variational principle to matrix product states

2019

— McArdle and co-workers run variational imaginary-time evolution on quantum hardware

Going further

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information, Ch. 2–4 (state evolution, dynamics) · Preskill, Lecture Notes on Quantum Computation, Ch. 4–5 · Watrous, The Theory of Quantum Information (distances and channels) · Kitaev, Shen & Vyalyi, Classical and Quantum Computation · Mermin, Quantum Computer Science.

Primary sources. McLachlan, A variational solution of the time-dependent Schrödinger equation, Mol. Phys. 8, 39 (1964) · Li & Benjamin, PRX 7, 021050 (2017) · McArdle et al., Variational ansatz-based quantum simulation of imaginary time evolution, npj Quantum Inf. 5, 75 (2019) · Yuan et al., Theory of variational quantum simulation, Quantum 3, 191 (2019) · Motta et al., Determining eigenstates and thermal states with quantum imaginary time evolution, Nat. Phys. 16, 205 (2020) · Stokes et al., Quantum natural gradient, Quantum 4, 269 (2020).

Provenance. The equations of motion follow the standard McLachlan/Dirac–Frenkel derivations; the geometric tensor and Fubini–Study metric are textbook gauge-invariant objects. A verifier should: (1) re-derive $A\dot{\boldsymbol\theta}=C$ by minimizing $\langle R|R\rangle$ ; (2) confirm $C^{\mathrm{(I)}}_i=-\tfrac12\partial_i\langle H\rangle$ from Hermiticity of $H$ ; (3) check monotone descent from $A\succeq0$ ; (4) verify $\dot\theta=\sin\theta$ and $\langle H\rangle(\tau)=-\tfrac12\tanh\tau\to-\tfrac12$ against normalized $\mathrm{e}^{-H\tau}|+\rangle$ ; (5) confirm $A=\tfrac14\begin{pmatrix}1&1\\1&1\end{pmatrix}$ has eigenvalues $\{\tfrac12,0\}$ . All numbers are exact algebraic values; no empirical constants are fitted.

Builds on: Variational Quantum Eigensolver · Parameterized Quantum Circuits and Ansatz Design · Parameter-Shift Rules and Quantum Gradients · Cost-Function and Measurement Estimation in Variational Algorithms Leads to: Classical Optimizers for Variational Quantum Algorithms · Barren Plateaus in Variational Landscapes · Strategies to Mitigate Barren Plateaus · Expressibility, Entangling Capability, and Trainability Trade-offs

Ask the tutor: Ask me to walk through the single-qubit VITE flow step by step, or to show exactly how the Fubini–Study metric turns plain gradient descent into the quantum natural gradient.

Imaginary-time evolution is not unitary, so how does a quantum computer implement it?

Correct — $\mathrm{e}^{-H\tau}$ is not a gate, and a quantum computer applies only unitaries. The variational trick dodges this: you never apply $\mathrm{e}^{-H\tau}$ to the state. You keep the state on a manifold of unitarily preparable circuits and move its classical parameters $\boldsymbol\theta$ so the trajectory mimics what $\mathrm{e}^{-H\tau}$ would have done — the non-unitary evolution emulated by ordinary unitary circuits with slowly changing angles. The related QITE algorithm emulates it differently, fitting a local unitary at each step to reproduce the update on a bounded correlation length.

How do the McLachlan, Dirac–Frenkel, and TDVP principles relate?

All three project the exact velocity onto the tangent space, and for real parameters they give the identical equation of motion $A\dot{\boldsymbol\theta}=C$ . They part company for complex or holomorphic parameterizations: Dirac–Frenkel imposes $\langle\delta\psi|(\mathrm{i}\partial_t-H)|\psi\rangle=0$ on complex-analytic parameters, while McLachlan minimizes the real residual norm over real velocities. The discrepancy lives in how each treats the imaginary part of the geometric tensor. Because NISQ ansätze use real rotation angles, McLachlan's form is the workhorse — the one that reduces cleanly to a measurable linear system.

Is VITE actually better than just running a classical optimizer on VQE?

Sometimes, for a specific reason: VITE is quantum natural-gradient descent, rescaling each step by the inverse metric. On ill-conditioned landscapes, where flat and steep directions coexist and plain gradient descent zig-zags, the preconditioner straightens the path and can converge in far fewer iterations, invariant to how the circuit is parameterized. The catch is cost: each step needs the full $O(p^2)$ metric plus a regularized solve, versus $O(p)$ for plain descent. Whether faster convergence pays for pricier steps depends on the problem, conditioning, and shot budget.

Why is the metric A so often singular?

Because parameterized circuits are riddled with redundancy. Two parameters whose generators commute on the same support, a symmetry-locked parameter, or a gate that leaves the state untouched all produce linearly dependent tangent vectors — and linear dependence forces an exact zero eigenvalue of $A=\mathrm{Re}\,G$ . Even without exact redundancy, deep circuits push many eigenvalues toward the shot-noise floor. So the rank of $A$ is usually well below $p$ , and every practical implementation solves $(A+\epsilon\mathbb{1})\dot{\boldsymbol\theta}=C$ or uses a truncated pseudo-inverse rather than $A^{-1}$ .

What exactly is the McLachlan residual, and why does it set the error?

It is the vector $|R\rangle=\sum_j\dot\theta_j|\partial_j\psi\rangle+(H-\langle H\rangle)|\psi\rangle$ — the mismatch between the velocity the manifold can supply and the one exact imaginary-time evolution demands. At the optimal $\dot{\boldsymbol\theta}$ , its squared norm $\langle R|R\rangle$ is the part of the exact velocity pointing out of the tangent space, which cannot be represented. This is the irreducible error even with perfect circuits — a dynamical measure of limited expressibility. It grows as the true state develops entanglement the circuit cannot reach, which is why variational trajectories eventually drift from exact ones.

Do barren plateaus affect VITE the way they affect VQE?

Yes — VITE does not rescue you from barren plateaus. On an exponentially flat landscape the force entries have variance shrinking like $2^{-n}$ , vanishing into the shot noise, while the metric loses spectral weight with $\lambda_{\min}(A)\to0$ . Both sides of $A\dot{\boldsymbol\theta}=C$ collapse together and the regularized solve returns essentially noise. This is why the topic points forward to barren-plateau mitigation: the preconditioner reshapes the landscape but cannot create signal where the plateau has destroyed it.