Warped phase transformation

What it fills

• Recast a non-Hermitian generator as Hamiltonian evolution

  Given a generator $A(t)$ 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.

When it applies

Unlike the LCHS identity there is no positive-semidefiniteness requirement in order to form the Schrödinger system, and that is the structural difference between the two ways through this slot. What it does require is the extra dimension: the auxiliary variable is continuous, so it has to be truncated and discretized, and the recovery is not free. The original solution is read back 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 $p* > 0$, and that $e^{p*}$ is the factor the non-unitarity was moved into.

Requires

Every step this method names moves its route along, so there is nothing it needs alongside them.

Example

Example 2.1 of the technical companion is the one run either paper reports, and the paper frames it as justification rather than as a quantum execution: "We conduct a numerical test in one dimension to justify the above idea." The test problem is the heat equation $u_t - u_{xx} = 0$ on $x \in (-1,1)$ with the periodic conditions $u(t,-1) = u(t,1)$ and $u_x(t,-1) = u_x(t,1)$, whose exact solution $u(t,x) = e^{-\pi^2 t}\sin(\pi x)$ is known and is what the numbers are compared against. The transformation is applied as this record states it, with one change made for the numerics: the extension to $p < 0$ is given its own rate, $w(0,x,p) = e^{-\alpha|p|}u_0(x)$, where $\alpha = 1$ is forced on $p > 0$ to match the exact solution and a large $\alpha$ is chosen on $p < 0$ so that the initial data occupies a small part of the truncated domain. The grids are $M = 2^4 = 16$ points in $x$ and $N = 2^9 = 512$ in $p$, and the time horizon is not free: the modes $\mu_{\pm 1}$ contribute most to the propagation, their wave speed is $s_1 = \pi^2$, and the wave must not reach the artificial boundary, so $t \le T_* = (L_0 - L)/s_1$. The first setting takes $L = -5$, $R = 5$, $\alpha = 10$ on $p < 0$ and $L_0 = -1$, giving $T_* = 0.4053$; the second reaches $t = T = 1$ by enlarging the domain to $L = L_0 - Ts_1$, about 11 in absolute value, with $\alpha = 40$ and $R = 10$. Both recovery formulas are exercised — the integral $u(t,x) = ∫_0^∞ w(t,x,p)\,dp$ and the pointwise $u(t,x) = e^{p_*}w(t,x,p_*)$, with $p_*$ taken at a grid point just to the right of $p = 0$ because "it is better to pick the point $p_* > 0$ near $p = 0$ ... to avoid the loss of significant digits of $e^{-p_*}$ for large $p_*$". The same settings are then run again with a finite difference and upwind scheme in place of the spectral one. What comes out is reported graphically and only graphically: numerical and exact solutions plotted together, and snapshots showing the wave has almost moved to the left end of the $p$ domain, "which validates the previous arguments". No error norm, no convergence rate and no table appears anywhere in the paper. And it is a classical computation throughout — the phase space equation is solved on a classical computer, and no quantum device or simulator is named.

Classical simulation · Example 2.1 of arXiv:2212.14703

given  the linear equation for u and its initial data u_0
#   worked here on the heat equation  d_t u - Laplacian u = 0

introduce one auxiliary variable p > 0
set     w(t,x,p) = e^{-p} u(t,x)
#   then  d_t w + d_p Laplacian_x w = 0   for p > 0

extend the initial data symmetrically to p < 0:  w(0,x,p) = e^{-|p|} u_0(x)
#   the wave moves from right to left, so no boundary condition is needed at p = 0,
#   and the extension does not impact the region p > 0

truncate the p-line:  impose w(t,x,p=-L) = w(t,x,p=L) = 0 for L > 0 large enough
#   justified by the exponential decay of e^{-|p|}
discretise p with step Dp and Fourier transform in p
#   what comes out is a Hamiltonian system; simulating it is the layer below

# recovery, afterwards:
#   u(t,x) = integral_0^inf w(t,x,p) dp
#   or     u(t,x) = e^{p*} w(t,x,p*) for any chosen p* > 0

Cost, as the source states it

Jin, Liu and Yu present the transformation and worked examples — the heat, convection, Fokker-Planck, linear Boltzmann and Black-Scholes equations, with extensions to Vlasov-Fokker-Planck and to the Liouville representation of nonlinear ODEs — rather than a single unified query-complexity theorem. So no like-for-like count against the LCHS figures is given here, and the absence is the paper's shape rather than an omission at this desk.

Implementations

• Fourier spectral discretisation of the phase space heat equation

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

  The scheme Example 2.1 actually runs. §2.1.1 turns the heat equation into what the paper calls the phase space heat equation, $\partial_t w + \Delta_x\partial_p w = 0$ on $p \in (-\infty,\infty)$ with $w(0,x,p) = e^{-|p|}u_0(x)$, and this entry is how that equation is put on a grid and solved: a Fourier spectral discretisation in both the spatial and the auxiliary variable, on the periodic one-dimensional problem whose exact solution is known.

  Uniform meshes $\Delta x = 2/M$ and $\Delta p = (R-L)/N$. The discrete Fourier transform in $x$ gives $\partial_t w(t,p) - P_\mu^2\partial_p w(t,p) = 0$, where $P_\mu$ is the matrix representation of the momentum operator $-i\partial_x$; it is diagonalised as $D_\mu = \Phi^{-1}P_\mu\Phi$ with $D_\mu = \mathrm{diag}(\mu_{-M/2},\dots,\mu_{M/2-1})$ and $\mu_l = \pi l$, so each component $\hat w_l$ is a linear hyperbolic equation whose wave "moves from the right to the left with speed $s_l = \mu_l^2$" and whose analytic solution $\hat w_l(t,p) = e^{-\alpha(p+s_lt)|p+s_lt|}\hat u_{0,l}$ is available to check against. Fully discretised, the grid values are straightened into $w(t) = \Sigma_{ij} w(t,x_i,p_j)|i,j\rangle$ and the transform on both variables gives $\partial_t w - i(P_\mu^2 \otimes P_\mu)w = 0$, then $\partial_t c - i(D_\mu^2 \otimes D_\mu)c = 0$ with $c = (F_x^{-1}\otimes F_p^{-1})w$. The tensor products are never formed: the paper uses the matrix identity $C = F_p^{-1}WF_x^{-T}$, "which avoids the use of memory-consuming tensor products".

  Initial data $u_0(x) = \sin(\pi x)$, chosen so that the exact solution $u(t,x) = e^{-\pi^2 t}\sin(\pi x)$ is available; its DFT coefficients are plotted, and "the amplitudes decay very fast in the Fourier mode $\mu_l$, and $\mu_{\pm 1}$ contribute most to the propagation". Grids $M = 2^4 = 16$ and $N = 2^9 = 512$. Two parameter settings: $L = -5$, $R = 5$, $\alpha = 10$ on $p < 0$, $L_0 = -1$ and $t = T_* = 0.4053$; then $t = T = 1$ with $L = L_0 - Ts_1$ (about 11 in absolute value), $\alpha = 40$ and $R = 10$. Nothing is drawn from outside the paper — every input is generated from the closed-form initial condition.

  The recovered $u(t)$ is plotted against the exact solution for both recovery formulas, the pointwise $u = e^{p_*}w(t,x,p_*)$ and the integral $u = ∫_0^∞ w\,dp$, at $t = T_*$ and again at $t = T = 1$. Snapshots of the modulus $|\hat w_{l^*}(t,p_j)|$ in the frequency space and of $w_{i^*}(t,p_j)$ in the original space show that "the waves in both spaces have almost moved to the left end, which validates the previous arguments". Agreement is reported graphically; no error norm, convergence rate or table is quoted. The run is classical, and no quantum device or simulator is named.

• Finite difference and upwind discretisation of the phase space heat equation

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

  The paper's second and independent check of the same construction: "One can also use the finite difference discretisation to further validate the above arguments." Same equation and same settings as the spectral run, different discretisation, so what the two share is the transformation rather than the numerical method.

  Central differences in $x$ with ghost points $x_{-1}$ and $x_{M+1}$, closed by assuming the discretisation is valid at $x = x_0$ and $x = x_M$ and then eliminating the ghost values by periodicity, give $\partial_t w(t,p) + A\partial_p w(t,p) = 0$ with $A$ the circulant tridiagonal $(1,-2,1)/\Delta x^2$; its eigenvalues are $\lambda_k(A) = -\frac{4}{\Delta x^2}\sin^2\frac{k\pi}{M}$. Since $\lambda_k \le 0$ the waves travel one way, so $p$ is discretised upwind and $t$ explicitly, $\frac{w_j^{n+1}-w_j^n}{\Delta t} + A\frac{w_{j+1}^n - w_j^n}{\Delta p} = 0$, closed at $j = N$ by the periodicity in $p$. The result is the iteration $W^{n+1} = BW^n$, with $B$ block bidiagonal in $I + A_1$ and $-A_1$ and $A_1 = \frac{\Delta t}{\Delta p}A$.

  The same settings as the spectral run; the paper says so rather than restating the numbers.

  "The numerical results are similar to that of the spectral method under the same settings", shown for both recovery formulas at $t = T_*$. Graphical agreement again, with no error norm quoted, and classical again.

• Applied after spatial discretisation, to the ODE system

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

  The same transformation in the other order, and the paper draws the distinction itself: "here we first discretise in space (hence $A$ is the difference matrix) and then use the warped phase transformation later, but in Section 2 we use the warped phase transformation first and then discretise in space". The problem statement is the ODE system $du(t)/dt = Au(t)$ with $A$ time-independent and in general non-Hermitian, which is what any linear PDE becomes after spatial discretisation; the short paper writes the same construction in the sign convention $du(t)/dt = -Au(t)$.

  $A$ is split into Hermitian parts, $A = H_1 + iH_2$, and "in view of the stability, it is natural to assume that $H_1$ is negative semi-definite" (the short paper writes $A = H + i\bar H$ with $H$ positive semi-definite under its opposite sign). The transformation is then applied to the vector $u(t)$ rather than to $u(t,x)$: $v(t,p) = e^{-p}u(t)$ for $p > 0$, with the initial data symmetrically extended as in §2.1.1, which turns the ODEs into a system of linear convection equations $\frac{d}{dt}v(t,p) = -H_1\partial_p v + iH_2 v$ with $v(0,p) = e^{-|p|}u_0$, recovered by $u(t) = ∫_0^∞ v(t,p)\,dp$. The discrete Fourier transform in $p$ gives the Hamiltonian system $\frac{d}{dt}w = -i(H_1 \otimes P_\mu)w + i(H_2 \otimes I)w$, and the change of variables $\tilde w = (I_u \otimes F_p^{-1})w$ replaces $P_\mu$ by the diagonal $D_\mu$. The paper then checks the two orderings against each other: for the heat equation the system "coincides with (2.22) since $H_1 = A$ and $H_2 = O$", so "the two treatments are equivalent due to the fact that $A$ is a Hermitian matrix", while for the convection equation $H_1 = O$ and "there is no need to apply the warped phase transformation since it is already a Hamiltonian system".

  No numerics are reported for this ordering. What the paper states for it is the general gate complexity of Theorem 3.1, which is quoted on the Schrödingerisation record's cost.

What it needs

Nothing below this — it bottoms out here.

Other ways to fill the same slot

Different approaches

• Kernel-weighted combination of unitary propagators

  Split $A(t)$ into its Hermitian and anti-Hermitian parts, $A = L + iH$, and write the non-unitary propagator as a kernel-weighted integral over the unitary propagators generated by the one-parameter family $kL(t) + H(t)$. 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.

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.

Sources

• Quantum simulation of partial differential equations via Schrodingerisation↗Shi Jin, Nana Liu, Yue Yu · 2022theory yes · simulation no · hardware no
• Quantum simulation of partial differential equations via Schrodingerisation: technical details↗Shi Jin, Nana Liu, Yue Yu · 2022theory yes · simulation yes · hardware no