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SlotLayer 2

Recast a non-Hermitian generator as Hamiltonian evolution

Given a generator A(t)A(t) whose evolution is not unitary, produce a Hermitian generator — or a quadrature-indexed family of them — on a space at least as large, whose unitary evolution reproduces the original dynamics, together with the map that recovers the original solution. Both halves are required: a construction that reaches a Hamiltonian and cannot get back is not a route.

Takes

The generator A(t)A(t) with no Hermiticity assumed, the interval [0,T][0,T], and an error tolerance ε\varepsilon.

Returns

A Hermitian generator or a family of them, the enlargement of the space that carrying them cost, and the map that reads the original solution back — with the weight that map applies stated, because that weight is where the non-unitarity was moved to rather than removed.

This one, drawn

Recast a non-Hermitian generator as Hamiltonian evolutionKernel-weighted combination of unitary propagatorsWarped phase transformationKernel-weighted combination of unitary propagators — click the name to read about itLCHS identityWarped phase transformation — click the name to read about itWarped phase transformationLinear ODE system — you start hereHamiltonian surrogate, with the map back — you finish here

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From Linear ODE system to Hamiltonian surrogate, with the map back

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Why this is a layer

Every route that reaches a simulator without forming a linear system has to pass through here, and the two ways through it demand different things of AA — a precondition, not a constant. LCHS requires the Hermitian part L(t) = (A(t)+A(t)^†)/2 to be positive semi-definite across the whole interval, and buys unitarity with a quadrature: the propagator becomes a kernel-weighted combination of unitary evolutions, and how fast the kernel decays is how many of them there are. The warped phase transformation requires nothing of the spectrum and buys unitarity with a dimension: one extra variable turns the system into a Schrödinger equation, and the price reappears at recovery, where the answer is read back out of that variable under a factor that grows with the decay being undone. Neither is a special case of the other. A cost model that says "reduce to Hamiltonian simulation" without saying which of these it used has not stated its precondition, and the precondition is the part that decides whether the route applies at all. This is also the slot that says which routes do *not* need it: the Koopman-von Neumann lift arrives holding a generator that is already Hermitian, so it goes straight to the simulator and this layer is not on its path.

Ways to do this

2 methods recorded

  • Kernel-weighted combination of unitary propagators

    Split A(t)A(t) into its Hermitian and anti-Hermitian parts, A=L+iHA = L + iH, and write the non-unitary propagator as a kernel-weighted integral over the unitary propagators generated by the one-parameter family kL(t)+H(t)kL(t) + H(t). Every member of that family is Hermitian by construction, so each is an ordinary Hamiltonian simulation problem and the combination is an LCU over them.

  • Warped phase transformation

    Introduce one extra variable and change to it, so that a linear ODE or PDE system becomes a system of Schrödinger equations in real time — which a simulator runs as it stands. The original solution lives in the auxiliary dimension and is recovered from it afterwards.

Routes that skip this layer

No recorded route avoids this step.

This is a step inside

  • LCHS — linear combination of Hamiltonian simulation

    Express a general non-unitary evolution operator as a linear combination of unitary evolution operators, each of which solves a Hamiltonian simulation problem, rather than converting the problem into a dilated linear system. An, Liu and Lin state that the method can achieve optimal cost in terms of state preparation.

  • LCHS with the improved kernel a narrower version of LCHS — linear combination of Hamiltonian simulation

    A family of identities expressing non-unitary evolution as a linear combination of unitary evolutions, built on the kernel f(z)=1/(Cβe(1+iz)β)f(z) = 1/(C_β e^{(1+iz)^β}) with β∈(0,1)β ∈ (0,1) and Cβ=2πe−2βC_β = 2π e^{-2^β}. The kernel decays at a near-exponential rate e−c∣k∣βe^{-c|k|^β}, replacing the original Cauchy kernel's quadratic decay and exponentially enhancing accuracy.

  • Schrödingerisation (linear PDEs as Schrödinger equations)

    A simple change of variable — the warped phase transformation, which introduces one extra variable — recasts any linear PDE or ODE system into a system of Schrödinger equations in real time, which ordinary Hamiltonian simulation then runs. The original solution is recovered from the auxiliary dimension.

In the Atlas

No record in the Atlas covers this yet. The catalogue is circuits and primitives; this part of the literature is not in it.