Matrix product state loading

• Prepare an input state

  Map $|0\ldots0⟩$ to a state whose amplitudes are proportional to a specified vector $b$, to within $\varepsilon$. The cost is set by which description of $b$ you hold, not by the algorithm that consumes it.

For functions that a matrix product state of small bond dimension approximates well. Holmes and Matsuura restrict to smooth, differentiable, real-valued functions with bounded derivatives, and in their small-system numerics on Gaussian, lognormal and Lorentzian distributions the fidelity stays above 99% down to a width σ = 0.1 but starts to fall once the width drops below about 0.12, as the distribution is squeezed (Fig. 5); their argument that the method scales rests on a conjecture, supported numerically, that bond dimension 2 stays a good approximation as qubits are added (Sec. VII C). Bohun et al. prove, for smooth functions real or complex, that the entanglement entropy across bond $k$ decays as $O(k/4^k)$ (Corollary 2), which covers the fine scales; across the coarsest bonds the behaviour is not universal and depends on how localised the function is.

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

Hardware, on IBM devices. Bohun et al. ran their circuits for Lévy, log-normal and Gamma distributions on ibm_torino at up to 25 qubits with 5000 shots, showing the better of their one- and two-layer circuits, and every one passed a Kolmogorov–Smirnov test at the 0.05 level with 200 samples (Fig. 9). They then built two-layer circuits by tensor cross interpolation and ran them on ibm_marrakesh at 50 qubits, where the samples no longer pass the statistical tests and the paper claims only that the qualitative features are reproduced (Fig. 10), and on ibm_kingston with 2000 shots and dynamical decoupling, for Gamma distributions on 100 qubits and a Lévy distribution on 156 (Fig. 11). So the 156-qubit figure is a hardware run, not a simulation, but no statistical test is reported for it: the paper's conclusion claims qualitative agreement up to 156 qubits and quantitative agreement up to 25.

Run on quantum hardware · Sec. V, Figs. 9 to 11 of arXiv:2412.05202

given  a smooth function f on [0, 1], N qubits, a number of layers,
       and a cutoff eps_trunc

sample f at the 2^N grid points x = 0.s1 s2 ... sN
    # past about 30 qubits, build the MPS by tensor cross interpolation
    # from calls to f instead of storing the 2^N samples
factorise the samples into a matrix product state by successive SVDs

circuit = empty
repeat for each layer:
    truncate every bond of the current MPS to its two largest singular values
    convert that bond-dimension-2 MPS exactly into one layer of two-qubit gates,
        starting from a chosen central bond, and leave out the gate on any bond
        whose discarded weight is already below eps_trunc
    apply the inverse of the layer to the untruncated MPS   # now less entangled
    put the layer in front of circuit                      # it will run first

# circuit applied to |0...0> approximates the state encoding f
# one layer's fidelity is about 1 - (sum over bonds of the discarded squared
# singular values), Eq. (20); Bohun et al. give no bound on the layers a given
# accuracy needs

Holmes and Matsuura (Sec. V D) turn a bond-dimension-2 matrix product state on $N$ qubits into $N + 1$ unitaries, $N$ of them two-qubit gates, in a serial circuit of depth $N + 1$, and state that standard decompositions give $7N + 1$ one-qubit and $3N$ two-qubit gates at a depth of about $6N$. Their classical construction, Algorithm 2, takes $O(N\chi^3)$ time, which they write as $O(8N)$ for $\chi = 2$ (Sec. VI). Bohun et al. build each disentangling layer from at most $N - 1$ two-qubit gates, with depth as low as $N/2 - 1$ when the layer starts from a central bond, and synthesise each two-qubit isometry with 2 CNOTs (Sec. IV); their two-layer circuits on 15 qubits average between 27.0 and 32.7 CNOTs across the distributions and cutoffs of Tables I and II. Bohun et al. do not bound how many layers a given accuracy needs: for one layer they estimate the fidelity from the discarded singular values, $F \approx 1 - \sum_{k>0,\, i\ge 2}\Lambda_{k,i}^{2}$ (Eq. 20), where $\Lambda_{k,i}$ is the $i$-th singular value at bond $k$, counted from $i = 0$, and for more layers they report numerics only. Holmes and Matsuura extract one circuit from one compressed state, and the accuracies they report are numerical (Figs. 5 to 8).

Neither cited paper links code for its method. Holmes and Matsuura (arXiv:2005.04351) point to no repository, and Bohun et al. (arXiv:2412.05202) name only the tools they built on, the Qiskit matrix-product-state simulator and the torchTT package, not an implementation of their algorithm.

Nothing below this — it bottoms out here.

Different approaches

• Uniformly controlled rotations

  Prepare an arbitrary state with one layer of uniformly controlled (multiplexed) Ry and Rz rotations per qubit, the angles computed analytically from the amplitude list. This is the exact, assumption-free method most software stacks emit by default.

• Grover-Rudolph bisection preparation

  Prepare a discrete approximation to a probability density by recursive bisection: at layer $k$ a uniformly controlled rotation splits each current interval's probability mass between its two halves, so only $n$ rotation layers are needed.

• Sparse state preparation

  When only $d$ of the $2^n$ amplitudes are nonzero, build the $d$ computational-basis strings directly instead of rotating through the whole binary tree, so the cost tracks $d$ and $n$ rather than $2^n$.

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

• Efficient Quantum Circuits for Accurate State Preparation of Smooth, Differentiable Functions↗Adam Holmes, A. Y. Matsuura · 2020theory yes · simulation ? · hardware no
• Entanglement scaling in matrix product state representation of smooth functions and their shallow quantum circuit approximations↗Vladyslav Bohun, Illia Lukin, Mykola Luhanko, Georgios Korpas, Philippe J. S. De Brouwer, Mykola Maksymenko, Maciej Koch-Janusz · 2024theory yes · simulation ? · hardware yes