Quantum Maps and Discrete-Time Evolution

A quantum map is a rule that advances a quantum system by one step. For a closed system, the basic discrete-time map is unitary:

$$\lvert\psi_{n+1}\rangle = U_n\lvert\psi_n\rangle.$$

If the same step is repeated,

$$\lvert\psi_n\rangle = U^n\lvert\psi_0\rangle.$$

For density operators,

$$\rho_{n+1} = U_n\rho_nU_n^\dagger.$$

This page treats closed-system discrete-time evolution. General noisy maps, measurement branches, and channels belong to Quantum Operations and Channel Composition and Fixed Points.

Continuous Versus Discrete Time

Continuous closed-system dynamics is usually written as

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle = H(t)\lvert\psi(t)\rangle.$$

Discrete-time dynamics instead specifies snapshots:

$$\lvert\psi_0\rangle, \lvert\psi_1\rangle, \lvert\psi_2\rangle, \ldots$$

and a rule connecting one snapshot to the next. The step may come from:

• observing a continuous system stroboscopically;
• one period of a periodically driven Hamiltonian;
• a kicked system with abrupt pulses;
• a product-formula time step;
• an ideal quantum circuit layer;
• an abstract unitary map used as a model.

The map is the primitive object. A Hamiltonian may generate it, but it is not always unique.

Fixed Unitary Step

For a fixed unitary $U$,

$$\lvert\psi_{n+1}\rangle = U\lvert\psi_n\rangle, \qquad n\in\mathbb Z_{\ge0}.$$

The solution is

$$\lvert\psi_n\rangle = U^n\lvert\psi_0\rangle.$$

The corresponding density-operator map is

$$\rho_n = U^n\rho_0(U^\dagger)^n.$$

The observable-side discrete Heisenberg map is

$$A_n = (U^\dagger)^n A_0 U^n.$$

Expectation values agree in the two descriptions:

$$\operatorname{Tr}(\rho_n A_0) = \operatorname{Tr}(\rho_0 A_n).$$

Spectrum of a Unitary Map

Because $U$ is unitary, its eigenvalues are phases. In finite dimension, choose an eigenbasis

$$U\lvert\phi_\alpha\rangle = e^{-i\theta_\alpha} \lvert\phi_\alpha\rangle, \qquad \theta_\alpha\in\mathbb R \quad \text{mod }2\pi.$$

If

$$\lvert\psi_0\rangle = \sum_\alpha c_\alpha\lvert\phi_\alpha\rangle,$$

then

$$\lvert\psi_n\rangle = \sum_\alpha c_\alpha e^{-in\theta_\alpha} \lvert\phi_\alpha\rangle.$$

Thus the spectrum of a closed unitary map gives oscillation phases, recurrences, and interference patterns. It does not give decay rates in a finite-dimensional closed system. Decay rates arise in nonunitary maps, coarse-grained descriptions, or limits with continuous spectra.

Effective Hamiltonian and Logarithm Branches

If the step represents a time interval $\tau$, one may try to write

$$U = e^{-iH_{\rm eff}\tau/\hbar}.$$

Formally,

$$H_{\rm eff} = \frac{i\hbar}{\tau}\log U.$$

The logarithm of a unitary is multi-valued. If

$$U = \sum_\alpha e^{-i\theta_\alpha} \lvert\phi_\alpha\rangle\langle\phi_\alpha\rvert,$$

then one branch gives

$$H_{\rm eff} = \sum_\alpha \frac{\hbar\theta_\alpha}{\tau} \lvert\phi_\alpha\rangle\langle\phi_\alpha\rvert.$$

Changing

$$\theta_\alpha \longrightarrow \theta_\alpha+2\pi m_\alpha, \qquad m_\alpha\in\mathbb Z,$$

changes the corresponding eigenvalue of $H_{\rm eff}$ by

$$\frac{2\pi\hbar m_\alpha}{\tau}.$$

Therefore a discrete unitary map does not determine a unique time-independent Hamiltonian unless extra branch, locality, smoothness, or physical-continuation information is supplied.

Floquet Maps

The same-phase sampling protocol, micromotion reconstruction, and aliasing consequences are developed in Stroboscopic Dynamics.

For a periodically driven closed system,

$$H(t+T)=H(t),$$

the one-period map is the Floquet operator

$$U_F(t_0) = U(t_0+T,t_0).$$

At integer periods,

$$\lvert\psi_n\rangle = U_F^n\lvert\psi_0\rangle.$$

The eigenphases of $U_F$ define quasienergies modulo $\hbar\Omega$, where $\Omega=2\pi/T$. The full Floquet theory, including reference phase and micromotion, is developed in Floquet Operators and Floquet Theorem in Quantum Mechanics.

This page only uses Floquet systems as a central example of a unitary quantum map.

Kicked Systems

A kicked system alternates smooth evolution with abrupt pulses. A schematic period might have a free evolution generated by $H_0$ and an instantaneous kick generated by $V_K$. The one-step map can take the form

$$U_F = e^{-iH_0T/\hbar} e^{-iV_K/\hbar},$$

depending on the convention for when the kick occurs. The rightmost factor acts first.

Kicked models are useful because a complicated time-dependent problem becomes a concrete unitary map. The spectrum and powers of that map control stroboscopic behavior. In quantum-chaos settings, this map-based formulation is often more natural than trying to assign a smooth Hamiltonian to every instant. Quantum Chaos Preview explains which spectral, transport, and operator diagnostics can be applied without mistaking unitary map evolution for classical state-space separation.

The warning is the same as for Floquet maps: the map tells what happens at selected times. It does not by itself describe all intra-step behavior unless the construction supplies it.

Kicked Rotor Preview develops the canonical example, including its standard map, exact kick–free factorization, dynamical localization, and quantum-resonance exceptions.

Quantum Circuits as Discrete Evolution

An ideal quantum circuit is also a discrete unitary map. If gates act in the order

$$U_1,\ U_2,\ \ldots,\ U_m,$$

then the circuit unitary is

$$U_{\rm circ} = U_m\cdots U_2U_1.$$

The rightmost gate acts first. A circuit diagram may display this order left-to-right or right-to-left depending on convention, so the basis and ordering rules must be declared.

Circuit evolution is not automatically a literal time discretization of a Hamiltonian. A gate may be an idealized unitary primitive, a compiled approximation to Hamiltonian evolution, or an abstract operation in an algorithm. The matrix conventions for common gates are collected in Quantum Gates and Quantum Gates Formula Card.

Product Formulas as Maps

Product formulas turn a continuous Hamiltonian problem into repeated discrete steps. For

$$H=H_A+H_B,$$

a first-order Trotter step is

$$S_1(\Delta t) = e^{-iH_A\Delta t/\hbar} e^{-iH_B\Delta t/\hbar}.$$

The approximate evolution after $N$ steps is

$$\lvert\psi_N\rangle = S_1(\Delta t)^N\lvert\psi_0\rangle, \qquad N\Delta t=t.$$

For finite $\Delta t$, $S_1$ is a unitary map when $H_A$ and $H_B$ are self-adjoint, but it is generally not the exact unitary $e^{-i(H_A+H_B)\Delta t/\hbar}$. The error analysis belongs to Trotter Product Formula and Baker–Campbell–Hausdorff Formula.

Fixed Points and Invariant Subspaces

For a unitary map, a state vector satisfying

$$U\lvert\phi\rangle = e^{-i\theta}\lvert\phi\rangle$$

returns to the same ray after one step. A density operator satisfying

$$U\rho U^\dagger = \rho$$

is stationary under the map. Equivalently,

$$[U,\rho]=0.$$

In the eigenbasis of $U$, density-matrix elements evolve as

$$\rho_{\alpha\beta}(n) = e^{-in(\theta_\alpha-\theta_\beta)} \rho_{\alpha\beta}(0).$$

Elements with $\theta_\alpha=\theta_\beta$ modulo $2\pi$ are stationary. Other coherences oscillate. In a finite-dimensional closed system, the unitary map does not drive generic states toward a unique fixed state.

Closed Maps Versus Channels

The closed-system map

$$\rho \longmapsto U\rho U^\dagger$$

is a special quantum channel with one Kraus operator. General channels can be nonunitary:

$$\Phi(\rho) = \sum_\mu K_\mu\rho K_\mu^\dagger.$$

Repeated channels,

$$\rho_n=\Phi^n(\rho_0),$$

can have convergence, fixed points, decay modes, and spectral radii inside the unit disk. Those are open-system and quantum-information questions, not closed-system unitary-map questions. The canonical treatment is Channel Composition and Fixed Points.

Common Mistakes

• Assuming a discrete map has a unique Hamiltonian generator.
• Forgetting that the rightmost factor acts first in a product of operators.
• Treating a Floquet effective Hamiltonian as if it describes all intra-period motion.
• Interpreting eigenphases of a unitary map as decay rates.
• Confusing an ideal circuit unitary with the noisy physical operation implemented by hardware.
• Applying channel fixed-point intuition to a finite-dimensional closed unitary map.
• Ignoring tensor-product ordering when translating circuit diagrams into matrices.

Cross-Links

• Operator Dynamics
• Time-Evolution Operator
• Floquet Operators
• Floquet Theorem in Quantum Mechanics
• Quantum Chaos Preview
• Trotter Product Formula
• Baker–Campbell–Hausdorff Formula
• Time Ordering
• Quantum Gates
• Quantum Gates Formula Card
• Quantum Operations
• Channel Composition and Fixed Points

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. H. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Physical Review 138, B979, 1965.
• F. Haake, Quantum Signatures of Chaos, 3rd ed., Springer, 2010.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.

Exercises

1. Prove that a repeated fixed unitary step gives $\lvert\psi_n\rangle=U^n\lvert\psi_0\rangle$.

Solution

For $n=1$, the statement is the definition:

$$\lvert\psi_1\rangle = U\lvert\psi_0\rangle.$$

Assume $\lvert\psi_n\rangle=U^n\lvert\psi_0\rangle$. Then

$$\lvert\psi_{n+1}\rangle = U\lvert\psi_n\rangle = U^{n+1}\lvert\psi_0\rangle.$$

The result follows by induction.

2. Let $U\lvert\phi_\alpha\rangle=e^{-i\theta_\alpha}\lvert\phi_\alpha\rangle$. Derive the evolution of $\rho_{\alpha\beta}(n)$.

Solution

In the eigenbasis of $U$,

$$\rho_n = U^n\rho_0(U^\dagger)^n.$$

Taking the $\alpha\beta$ matrix element gives

$$\rho_{\alpha\beta}(n) = e^{-in\theta_\alpha} \rho_{\alpha\beta}(0) e^{in\theta_\beta} = e^{-in(\theta_\alpha-\theta_\beta)} \rho_{\alpha\beta}(0).$$

3. If gates act as $U_1$ first and $U_2$ second, what is the total unitary?

Solution

The state first becomes

$$\lvert\psi_1\rangle = U_1\lvert\psi_0\rangle.$$

Then

$$\lvert\psi_2\rangle = U_2\lvert\psi_1\rangle = U_2U_1\lvert\psi_0\rangle.$$

Thus the total unitary is $U_2U_1$, with the first-acting operator on the right.

4. Show that $U=-I$ does not determine a unique effective Hamiltonian for a one-dimensional Hilbert space and step time $\tau$.

Solution

One wants

$$-1 = e^{-iE_{\rm eff}\tau/\hbar}.$$

This holds when

$$\frac{E_{\rm eff}\tau}{\hbar} = (2m+1)\pi, \qquad m\in\mathbb Z.$$

Therefore

$$E_{\rm eff} = \frac{(2m+1)\pi\hbar}{\tau}.$$

The different integers are different logarithm branches of the same unitary phase.

5. Prove that a unitary map preserves purity.

Solution

For one step,

$$\rho' = U\rho U^\dagger.$$

Then

$$(\rho')^2 = U\rho U^\dagger U\rho U^\dagger = U\rho^2U^\dagger.$$

Taking the trace,

$$\operatorname{Tr}(\rho')^2 = \operatorname{Tr}(U\rho^2U^\dagger) = \operatorname{Tr}\rho^2$$

by cyclicity.