Stated for general linear partial differential equations: unlike LCHS there is no positive-semidefiniteness requirement in order to form the Schrödinger system, and that is the structural difference between the two. The cost reappears at recovery: the original solution is read back from the warped variable, either as $u(t,x) = ∫_0^∞ w(t,x,p) dp$ or pointwise as $u(t,x) = e^{p*} w(t,x,p*)$ for a chosen $p* > 0$. Worked examples include the heat, convection, Fokker-Planck, linear Boltzmann and Black-Scholes equations, with extensions to the Vlasov-Fokker-Planck equation and to the Liouville representation equation for nonlinear ODEs — which is how a nonlinear problem reaches this method. The companion technical paper does prove gate-complexity theorems — Theorem 3.1 for the general method, recorded under cost — but in step sizes and qubit counts whose $\varepsilon$-dependence is left to the discretisation scheme, so no like-for-like count against the LCHS figures is given here.

A block-encoding of $A(t)$ with a normalization $\alpha_A \ge \max_t \lVert A(t)\rVert$, preparation unitaries for $u_0$ and $b$, the evolution time $T$, and an error tolerance $\varepsilon$.

1. Linear ODE system → Hamiltonian surrogate, with the map backRecast a non-Hermitian generator as Hamiltonian evolution

   For the spatially discretised system $du(t)/dt = -Au(t)$ the generator "is in general not Hermitian", so $A$ is split into Hermitian parts, $A = H + i\bar{H}$ with $H = (A+A^\dagger)/2$ and $\bar{H} = i(A^\dagger - A)/2$. assumption: To ensure stability, $H$ is assumed positive semi-definite — a stability assumption, not something needed in order to form the Schrödinger system, which is the difference this record already records against LCHS. The warped phase transformation is then applied to $u(t)$ in the $p > 0$ region, $v(t,p) = e^{-p}u(t)$, the initial condition is extended to $p<0$ by $v(0,p) = e^{-|p|}u(0)$, and on $p \in (-\infty,\infty)$ the new variable satisfies $\partial_t v + H\partial_p v - i\bar{H}v = 0$; Fourier transforming in $p$ gives $i\partial_t \tilde{v} = (\eta H + \bar{H})\tilde{v}$, "which is still a system of Schrödinger equations, one for each $\eta$, since $\eta H + \bar{H}$ is Hermitian". approximation: That continuous family is discretised in $\eta$ to a single Hermitian generator, $i\,d\tilde{v}/dt = (H \otimes D + \bar{H} \otimes 1)\tilde{v} = H^{total}\tilde{v}$, with $D = \mathrm{diag}(\mu_1,\dots,\mu_N)$, $\mu_j = \pi(j - N/2)$. What goes down with it is the recovery map this slot's contract requires: $u(t) = \int_{-\infty}^{\infty}\chi(p)v(t,p)\,dp$, with $\chi$ the indicator of $p \ge 0$.

   approximationassumption

   Open this step

2. Hamiltonian surrogate, with the map back → Circuit for e^{-iHt}Simulate Hamiltonian evolution

   What the recasting hands down is $H^{total}$, Hermitian, so this stretch is ordinary sparse-access Hamiltonian simulation: "an $s$-sparse Hamiltonian $H$ acting on $m_H$ qubits can be simulated within error $\varepsilon$ with $O(\tau \log(\tau/\varepsilon)/\log\log(\tau/\varepsilon))$ queries" and $O(\tau(m_H + \log^{2.5}(\tau/\varepsilon))\log(\tau/\varepsilon)/\log\log(\tau/\varepsilon))$ additional two-qubit gates, "where $\tau = s\|H\|_{max}t$ and $t$ is the evolution time". assumption: Sparse access to the matrix — the black boxes $O_A$ and $O_F$ of Definition 3.1 — and a generator that does not move: "in the sequel, we assume $A$ is independent of time". The extra variable is what this stretch is charged for, through the norm parameter rather than through the dimension: the proof of Theorem 3.1 uses $\|D_\mu\|_{max} \lesssim 1/\Delta p$, which is where its $\tilde{O}(s(A)\|A\|_{max}/\Delta p)$ comes from. approximation: In the special case where $H_1$ diagonalises in the momentum basis and $H_2$ is diagonal, the Hamiltonian system is not simulated by that route at all but solved by the first-order time splitting scheme, "as in the proof of Theorem 2.1".

   approximationassumption

   Open this step

3. Circuit for e^{-iHt} → Answer about the solutioninverse Fourier transform, keep p > 0

   "By applying an inverse quantum Fourier transform $F_p^{-1}$, with respect to p, onto the second register we obtain $|w(t)\rangle$", and the solution is then recovered by restricting to $p > 0$ — as the integral $u(t,x) = ∫_0^∞ w(t,x,p) dp$, or on the state by projecting onto $1 \otimes \Sigma_{k=N/2}^{N} |k⟩⟨k|$. assumption: Only the $p > 0$ half-line carries u, so what survives is what that projection keeps: "a simple projection retrieves |u(t)⟩ with probability (‖u(t)‖‖exp(−p)‖/‖w(t)‖)^2 ∼ N(‖u(t)‖/‖w(t)‖)^2". Amplitude amplification with the oracle $Q = −S_w S_p$ raises that to ∼√N‖u(t)‖/‖w(t)‖, at Õ(‖w(t)‖/(√N‖u(t)‖)) queries to Q. The paper also allows a pointwise recovery instead: choose any $p* > 0$ and take $u(t,x) = e^{p*} w(t,x,p*)$.

   assumption

A state proportional to $u(T)$, or a history state, together with separately stated matrix-query and initial-state-query complexity.

The short paper (arXiv:2212.13969) carries no numerical test and points elsewhere for one: "For more details and numerical experiments on justifying the setup in Eq. (14), see our technical companion paper". That companion's Example 2.1 is a classical spectral solve of the one-dimensional heat equation, and it tests the warped phase transformation — the recasting — rather than this route end to end. It is written up on the `warped-phase-transformation` record, where it belongs.

given  A from the spatial discretisation (in general non-Hermitian), u_0, evolution time t
#   du(t)/dt = -A u(t)

split A into Hermitian parts:
    A = H + i H_bar,   H = (A + A^dag)/2,   H_bar = i(A^dag - A)/2
#   to ensure stability, H is assumed positive semi-definite

# ---- hamiltonian-recasting, via the warped phase transformation ----
introduce p > 0 and set   v(t,p) = e^{-p} u(t)
extend the initial data:  v(0,p) = e^{-|p|} u(0)   on p in (-inf, inf)
#   v then satisfies  d_t v + H d_p v - i H_bar v = 0
Fourier transform in p:   i d_t v~ = (eta H + H_bar) v~
#   one Schrodinger equation for each eta, since eta H + H_bar is Hermitian
discretise eta:           i d/dt v~ = (H (x) D + H_bar (x) 1) v~ = H_total v~

# ---- hamiltonian-simulation ----
simulate the Hermitian H_total under sparse access to it
#   no linear system is assembled, and no time discretisation is chosen

# ---- this method's own stretch: read the answer back out of p ----
apply the inverse QFT F_p^{-1} on the p register to obtain |w(t)>
project onto p > 0:   1 (x) sum_{k=N/2}^{N} |k><k|
#   or pointwise instead: u(t,x) = e^{p*} w(t,x,p*) for a chosen p* > 0
amplitude-amplify with Q = -S_w S_p to recover |u(t)>

The companion technical paper states the general bound as Theorem 3.1: gate complexity $N_{Gates} = (m_d + m_p) \tilde{O}( s(A) ||A||_{max} / \Delta p ) + O( m_p \log m_p )$ for the spatially discretised linear system $\partial_t u = -A u$, where $s(A)$ is the sparsity of $A$, $m_d$ and $m_p$ count the system and auxiliary-register qubits, and $\Delta p$ is the auxiliary variable's grid step; when $H_1$ diagonalises in the momentum basis and $H_2$ is diagonal this improves to $T/\Delta t \cdot O(d m \log m + m_p \log m_p)$. Per-equation prices follow the same pattern — heat (Theorem 2.1), convection (Theorem 2.2), Black-Scholes (Theorem 4.1), linear Boltzmann (Theorem 4.3). All of these are deliberately stated in step sizes and qubit counts: the paper says the $\varepsilon$-dependence "is determined by the particular scheme one wishes to use", which is why no single $\varepsilon$-form figure is quoted here.

• Algorithm I — diagonal Hamiltonian evolution between QFTs, under first-order splitting

  • Quantum simulation of partial differential equations via Schrodingerisation: technical details

    Shi Jin, Nana Liu, Yue Yu · 2022

  About

  The paper's own name for the cheap route, fixed in Remark 2.4: "we refer to the algorithm mentioned in [21,27] to implement $e^{iHt}$ as Algorithm I, where $H$ is a diagonal matrix". It is what the paper uses whenever the Schrödingerised generator can be brought to diagonal form — the heat, convection and Black-Scholes equations, and the special case of the general theorem. The two references it names for the diagonal-unitary implementation are outside this record and are not cited here.

  Methods

  For the heat equation the Schrödingerised generator is $H = (P_1^2 + \dots + P_d^2 - V) \otimes D_\mu$, and it is solved by first-order time (Trotter) splitting: one step evolves the kinetic part, diagonal in the momentum basis as $H_D = ((D_1^\mu)^2 + \dots + (D_d^\mu)^2)\otimes D_\mu$, the second evolves the potential part $H_V = V \otimes D_\mu$, diagonal in the original basis, and a Fourier transform sits between them — $\tilde c^n \to e^{iH_D\Delta t} \to \tilde c^* \to (F_x \otimes I) \to \tilde w^* \to e^{-iH_V\Delta t} \to \tilde w^{n+1} \to (F_x^{-1}\otimes I) \to \tilde c^{n+1}$. Each one-dimensional quantum Fourier transform costs $O(m\log m)$ gates and each diagonal unitary $O(m_H)$ gates, with $m_H = m_d + m_p$, $m_d = dm$, $m \sim \log(1/\Delta x)$ and $m_p \sim \log(1/\Delta p)$. The QFT on the $p$ register is not inside the loop: it "is only performed twice".

  Data

  None found yet.

  Code

  None found yet.

  Results

  Gate counts, not measurements. Theorem 2.1 for the heat equation gives $N_{Gates} = T/\Delta t \cdot O(dm\log m + m_p\log m_p)$; Theorem 2.2 for the convection equation gives $N_{Gates} = O((d+1)m\log m)$; Theorem 4.1 for Black-Scholes gives $N_{Gates} = O(m\log m + m_p\log m_p)$; and the special case of Theorem 3.1, when $H_1$ diagonalises in the momentum basis and $H_2$ is diagonal, returns the same $T/\Delta t \cdot O(dm\log m + m_p\log m_p)$. Remark 3.3 adds that the splitting is unconditionally stable, so $\Delta t = O(\Delta x)$ may be taken and the heat equation's count becomes $T/\Delta x \cdot O(dm\log m + m_p\log m_p)$. The $\varepsilon$-dependence is left open deliberately — "for the given error bound $\varepsilon$, the $\varepsilon$-dependence of these quantities is determined by the particular scheme one wishes to use" — with the heat equation's mesh strategy given as $\Delta x \sim (\varepsilon/d)^{1/\ell}$, $\Delta t \sim \varepsilon$, $\Delta p \sim \varepsilon$, and the note that the initial condition's lack of regularity "implies first-order accuracy on $p$". No execution is reported: no device, no simulator, no counted circuit.

• Algorithm II — sparse access Hamiltonian simulation

  • Quantum simulation of partial differential equations via Schrodingerisation: technical details

    Shi Jin, Nana Liu, Yue Yu · 2022

  • Quantum simulation of partial differential equations via Schrodingerisation

    Shi Jin, Nana Liu, Yue Yu · 2022

  • Hamiltonian simulation with nearly optimal dependence on all parameters

    Dominic W. Berry, Andrew M. Childs, Robin Kothari · 2015

  About

  The general route, imported rather than built: Lemma 3.1 is labelled "Algorithm II, Theorems 1-2 in [5]", which is Berry, Childs and Kothari's near-optimal sparse Hamiltonian simulation. It is what the paper uses when the Schrödingerised generator cannot be brought to diagonal form, and it is also the route the short paper takes for the heat equation, where the same lemma carries its Theorem 3 and Theorem 4.

  Methods

  The Hamiltonian to be simulated is $-i(H_1 \otimes D_\mu) + i(H_2 \otimes I)$, reached from $du(t)/dt = Au(t)$ by splitting $A = H_1 + iH_2$, applying the warped phase transformation and Fourier transforming in $p$. Access to it is the sparse-access 4-tuple $(s, \|A\|_{max}, O_A, O_F)$ of Definition 3.1, with $O_A|j\rangle|k\rangle|z\rangle = |j\rangle|k\rangle|z \oplus A_{jk}\rangle$ and $O_F|j\rangle|l\rangle = |j\rangle|F(j,l)\rangle$, where $F$ returns the column index of the $l$-th non-zero element in row $j$. Lemma 3.1 then simulates an $s$-sparse $H$ on $m_H$ qubits within error $\varepsilon$ using $O(\tau\log(\tau/\varepsilon)/\log\log(\tau/\varepsilon))$ queries and $O(\tau(m_H + \log^{2.5}(\tau/\varepsilon))\log(\tau/\varepsilon)/\log\log(\tau/\varepsilon))$ additional two-qubit gates, with $\tau = s\|H\|_{max}t$. The generator is assumed not to move: "in the sequel, we assume $A$ is independent of time", with time-dependent Hamiltonians left to other references.

  Data

  None found yet.

  Code

  None found yet.

  Results

  Theorem 3.1's general bound, $N_{Gates,Schr} = (m_d + m_p)\tilde{O}(s(A)\|A\|_{max}/\Delta p) + O(m_p\log m_p)$, which this record already carries under cost, and Theorem 4.3 for the linear Boltzmann equation, whose count is assembled from two applications of the lemma. Where the auxiliary variable is charged is stated explicitly: $\|D_\mu\|_{max} \lesssim 1/\Delta p$ inside the proof, and the paper's own warning that for the convection equation of §2.2, whose $p$-derivative is second order, this "leads to $1/\Delta p^2$ as the multiplicative factor in the time complexity if Algorithm II is used". The short paper reaches the same shape from the other end, $\tilde{O}((\|u(0)\|/\|u(t)\|)st\|H\|_{max}/\varepsilon)$ in its Theorem 3. No execution is reported for any of it.