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Who can compete with quantum computers? Lecture notes on quantum inspired tensor networks computational techniques

Xavier Waintal, Chen-How Huang, Christoph W. Groth

arXiv:2601.03035v3quant-phcond-mat.str-el

TL;DR

Whether quantum-circuit states of interest remain classically simulable motivates these lectures. They develop tensor-network methods, and report that specified parameterized circuits span a subspace of dimension O(DN), much smaller than 2^N.

  • Problem

    Whether the subspaces explored by quantum circuits lie in relevant parts of Hilbert space and remain classically simulable is an open question.

    Source: §1 · p. 3, §2.3 · p. 8

  • Method

    The lectures present tensor networks as tools for large-scale linear algebra and develop quantics MPO constructions and TCI for representing and manipulating problems.

    Source: §3.2 · p. 11, §10.3 · p. 52, §11 · p. 61

  • Results

    For the specified parameterized circuits, the spanned subspace has dimension O(DN), much smaller than 2^N.

    Source: §2.3 · p. 8

  • Takeaways & Limitations

    The authors view tensor networks and TCI as likely to play a role in mapping problems into tensor-network form, while describing this prospect as uncertain.

    Source: §11 · p. 61

    The described MPS construction is impractical beyond small tensors because it requires all d^N entries and has exponential cost.

    Source: §4.3.1 · p. 16

Abstract

from arXiv · show

This is a set of lectures on tensor networks with a strong emphasis on the core algorithms involving Matrix Product States (MPS) and Matrix Product Operators (MPO). Compared to other presentations, particular care has been given to disentangle aspects of tensor networks from the quantum many-body problem: MPO/MPS algorithms are presented as a way to deal with linear algebra on extremely (exponentially) large matrices and vectors, regardless of any particular application. The lectures include well-known algorithms to find eigenvectors of MPOs (the celebrated DMRG), solve linear problems, and recent learning algorithms that allow one to map a known function into an MPS (the Tensor Cross Interpolation, or TCI, algorithm). The lectures end with a discussion of how to represent functions and perform calculus with tensor networks using the "quantics" representation. They include the detailed analytical construction of important MPOs such as those for differentiation, indefinite integration, convolution, and the quantum Fourier transform. Three concrete applications are discussed in detail: the simulation of a quantum computer (either exactly or with compression), the simulation of a quantum annealer, and techniques to solve partial differential equations (e.g. Poisson, diffusion, or Gross-Pitaevskii) within the "quantics" representation. The lectures have been designed to be accessible to a first-year PhD student and include detailed proofs of all statements.

1 Foreword

The lectures challenge the claim that quantum states are beyond classical computation by pointing to states calculated with classical algorithms and to the mathematical structures that make such predictions possible. They teach tensor-network methods through quantum-computing applications and extend them to large-scale linear algebra and PDEs.

1 Foreword

  • Classical algorithms have calculated many exponentially large quantum states, sometimes at machine precision, by exploiting mathematical structure [1, 2].The notes present this as the resolution of the apparent paradox that classical computers cannot store such states directly once N > 50.Source: §1 · p. 3
  • The lectures introduce tensor-network techniques through gate-based and annealing quantum-computing contexts, while focusing on core methods rather than much of the usual many-body formalism.The notes also offer subjective commentary intended to help readers assess claims of quantum advantage and notice what scientific articles leave unshown.Source: §1 · pp. 3–4

1. Foreword and 1.1 What’s in the lectures?

  • The lectures pair blackboard presentations with hands-on sessions to implement algorithms from scratch, and point readers to code repositories and existing implementations such as ITensor.The sessions used Python with NumPy, though Julia is also identified as a valid choice.Source: §1 · p. 4, §1.1 · p. 5

1.1 What’s in the lectures?

  • The curriculum covers MPS and MPO linear algebra, DMRG, TCI for converting index-based functions into MPS or MPO, and quantics methods for PDEs and Fourier transforms.It includes operations such as compression, matrix-vector products, and solving linear problems; in the quantics setting, Fourier transforms become MPO–MPS products described as exponentially faster than the regular FFT.Source: §1.1 · p. 4
  • The notes omit advanced network families, fermions and bosons, symmetries, TDVP, and several other tensor-network techniques.The examples use only qubits and spins.Source: §1.1 · pp. 4–5

1.2 Structure of the lectures

  • The notes progress from exact quantum-computer simulation and tensor-network concepts to compression, DMRG, MPO/MPS linear algebra, TCI, and quantics PDE solving.The later sections treat MPOs and MPSs as compact representations of very large matrices and vectors for linear algebra.Source: §1.2 · p. 5

2 The quantum computer: a machine for performing certain matrix-vector multiplications

The section presents gate-based quantum computing as matrix-vector operations on exponentially large state vectors, then describes limits on what circuits can explore, measure, and execute reliably.

2.1 An exponentially large internal state

  • The state of N qubits is a vector of 2^N complex values, too large to store classically once N is about 50 or more.Its components are indexed by all combinations of the N qubit values.Source: §2.1 · p. 6

2.2 Quantum circuits

  • A gate-based computer applies one- and two-qubit gates to evolve its state, then measurement returns probabilistic outcomes rather than the full state vector.The section frames these operations as a subset of linear algebra: matrix-vector multiplications using special gate matrices on exponentially large vectors.Source: §2.2 · pp. 6–7, §2.3 · p. 8

2.3 Summary

  • Only N bits of probabilistic information are obtained at the end, an output bottleneck that limits access to the exponentially large wavefunction.The lectures introduce classical techniques for exploring finite subspaces of the full Hilbert space to help put quantum-computing claims into perspective.Source: §2.3 · p. 8
  • Quantum circuits explore a subspace of dimension O(DN), far smaller than the full 2^N-dimensional Hilbert space.For 100 qubits and depth 10^6, the explored subspace still has 20 orders of magnitude fewer degrees of freedom than the full space; the passage also links barren plateaus to chaotic states in this subspace.Source: §2.3 · p. 8

2.4 Digression on decoherence and fidelity

  • The passage reports that state fidelity decreases exponentially, while finite operation accuracy—ε around 10^-3 for the best current quantum hardware—strongly limits practical circuit depth.Quantum error correction uses several physical qubits to build one higher-quality logical qubit, but its practical construction adds many layers of hardware complexity.Source: §2.4 · pp. 8–9

3 Tensor networks: basic notation and operations

The section introduces tensors, contractions, and factorization as tools for representing and operating on large objects through networks of smaller tensors. It emphasizes that contraction order affects computational cost, and shows how index fusion enables standard matrix factorizations.

3.1 Defining tensors

  • A tensor generalizes a vector or matrix to any number of indices, and its degree is the number of indices, each with a specified dimension.Diagrammatically, each index is represented by a leg.Source: §3 · p. 9, §3.1 · p. 9

3.2 Tensor contraction and tensor networks

  • Contraction identifies matching-dimension indices and sums over their values; tensor networks represent collections of tensors joined through such contracted indices.Uncontracted indices remain free and may serve as physical indices; a network's graph nodes carry tensors, and its edges correspond to indices.Source: §3.2 · pp. 10–11
  • CPU scaling falls from D^5 (D^21 for a 20-tensor MPS) to D^4 (remaining D^4 for 20 tensors) when contraction proceeds horizontally rather than vertically.Choosing a contraction order is generally NP-complete; computational time and memory footprint are key metrics, and one can sometimes trade one for the other.Source: §3.2 · pp. 11–12

3.3 Factorization

  • Fusing tensor indices turns a multi-leg tensor into a matrix, allowing standard linear-algebra factorizations such as SVD and LU to be applied.A matrix stored as a vector uses a composite index, with its ordering depending on row-major or column-major convention.Source: §3.3 · p. 12

3.4 Example: Factorizing the controlled NOT gate

  • The controlled-NOT gate is a four-legged tensor that factorizes into two three-legged tensors with internal rank 2, while a general two-qubit gate can require rank 4.Changing basis switches the roles of the control and target qubits.Source: §3.4 · pp. 13–14

4 Basic quantum computer emulators

This section surveys quantum-circuit emulation with tensor-network methods, comparing exact state-vector and MPS emulators with amplitude-based simulation in terms of costs, capabilities, and limitations. It also covers fast methods for restricted gate sets and introduces compression as central to practical tensor-network applications.

4.1 A quantum circuit is a tensor network.

  • Contracting a quantum circuit tensor network yields its many-qubit wavefunction, reducing classical emulation to a tensor-network contraction problem.The circuit’s one- and two-qubit gates have the same form as tensor contractions.Source: §4.1 · p. 14

4.2 Long and narrow quantum circuits: the full state simulator

  • The state-vector emulator stores 2^N complex amplitudes and runs in O(Ng2^N), making it suitable for deep, few-qubit circuits but exponentially costly in memory.At double precision, the notes estimate 16 GB for 30 qubits and 16 TB for 40 qubits; a laptop can simulate about 20–30 qubits this way.Source: §4.2 · pp. 14–15

4.3.2 MPS exact emulator: nearest neighbor gates

  • Exact MPS evolution keeps complexity linear in qubit count but exponential in circuit depth, as neighboring two-qubit gates can increase bond rank by up to four.A direct MPS decomposition requires access to all d^N state elements and has exponential cost in N; the initial product state has rank χ = 1, and one-qubit gates do not increase rank.Source: §4.3.2 · pp. 16–17, §4.3.1 · p. 16

4.4 Extension to arbitrary gates: introducing the MPO-MPS product

  • MPO-MPS products extend the emulator to distant gates; using SWAP gates instead is typically significantly more costly.The zip-up approach repeatedly contracts tensors and splits the result with QR or SVD to recover MPS form.Source: §4.4 · pp. 17–18
  • For a GHZ circuit, the full-state simulator scales exponentially with qubit count, while the exact MPS simulator scales linearly per gate with χ = 2.Figure 1 compares the two methods; χ = 2 is exact for this rank-2 MPS state.Source: §4.4 · p. 18, Figure 1 · p. 19

4.5.1 The tensor network route to observables

  • An MPS permits exact sampling and observable calculations; separately, the described tensor algorithm uses χ²d memory to retain its red tensor and has computing time scaling as Nχ³d.Sampling proceeds sequentially, drawing each bit conditioned on previously sampled bits; an unsuitable contraction sequence can cause catastrophic computing-time and memory costs.Source: §4.5.1 · p. 18, §4.5.2 · pp. 19–20

4.6 Amplitude simulations of a quantum computer

  • Weak simulation can obtain samples by computing only a few amplitudes, while strong simulation computes the full state; tensor-network contraction heuristics and slicing can help evaluate amplitudes.A two-sided MPS contraction is one strategy, and the notes state that amplitude computation is significantly harder than sampling alone.Source: §4.6 · pp. 21–22

4.7 Some remarks on “quantum supremacy”

  • Exact simulation costs generally grow exponentially with qubit count or circuit depth, but structured circuits can be easier than deliberately randomized quantum-supremacy circuits.The notes report that an initial estimate of 10 000 years for one simulation was revised down to 6 seconds, and describe such circuits’ outputs as chaotic and nearly indistinguishable from random-generator output.Source: §4.7 · pp. 22–23
  • Fast emulators can simulate circuits made only of Clifford gates, or mostly of them, for systems with thousands of qubits, even when the states are highly entangled.Source: §4.7 · p. 23
  • Quantum Monte Carlo is presented as a classical alternative to the quantum-computer-proposed variational quantum eigensolver (VQE) algorithm.Source: §4.7 · p. 23
  • Compression is introduced as the central idea behind most practical and useful tensor-network applications.Source: §4.7 · p. 23

5 Compressing many-body states with matrix product states

This section develops MPS compression as a low-rank approach to simulating quantum computers, covering its link to entanglement and algorithms from SVD truncation to TEBD and DMRG. It also explains how two-site DMRG grows MPS rank through joint optimization and controls it with truncated SVD.

5.1.1 A glimpse at the cross interpolation formula

  • Low-rank compression represents a matrix using smaller factors, and cross interpolation gives an exact reconstruction for exactly low-rank matrices from selected rows and columns.For approximately low-rank matrices, cross interpolation can serve as an approximation.Source: §5.1 · p. 23, §5.1.1 · p. 24

5.1.2 The Singular Value Decomposition

  • The best rank-χ approximation in Frobenius norm retains the χ largest singular values, making truncated SVD a central tensor-network compression method.The notes establish global optimality using the von Neumann trace inequality.Source: §5.1.2 · pp. 24–25

5.2 Entanglement entropy, area law and volume law

  • Entanglement determines the MPS matrix sizes required: maximal entanglement has entropy logχ, while area-law and volume-law scaling characterize different simulation challenges.For local, gapped one-dimensional Hamiltonians, ground-state entanglement entropy saturates at a finite value as N grows; volume-law states are described as particularly challenging for tensor networks.Source: §5.2 · p. 26

5.3 Canonical form

  • MPS canonical form places singular values at an orthogonal center, enabling local SVD compression that yields the best low-rank approximation of the overall state.The maximum entanglement representable by a rank-χ MPS is logχ, and optimal truncation requires canonical form before applying SVD.Source: §5.3 · pp. 26, 28, §5.3.1 · p. 29

5.4 Approximate TEBD for quantum circuits

  • Approximate TEBD restores MPS form after each two-qubit gate by truncating an SVD, with either a target bond dimension or discarded-weight tolerance controlling the approximation.In the random-circuit example, fidelity remains one until the bond-dimension cap is reached, then decreases exponentially with two-qubit gates; increasing bond dimension flattens the decay, and one-qubit gates do not reduce fidelity.Source: §5.4 · pp. 29–30

5.5 DMRG for quantum circuits (1 site)

  • Single-site DMRG for quantum circuits sweeps across MPS tensors, optimizing one tensor at a time to improve an approximate circuit output, typically starting from TEBD.The local optimization uses the canonical form and a fidelity objective; the optimized tensor also gives the current fidelity, and the fitting procedure can be adapted to MPO–MPS products.Source: §5.5 · pp. 31–32

5.6 DMRG for quantum circuits (2 sites)

  • Single-site DMRG fixes bond dimension and can become trapped in local minima; enrichment adds orthogonal vectors during canonicalization so the optimized center can reach rank 2χ.This enrichment is presented as one way to address the limitations of single-site DMRG.Source: §5.6 · p. 32
  • Two-site DMRG fuses neighboring tensors, optimizes the pair using the single-site formula with two holes, then splits it by truncated SVD to control rank.This is presented as the most common way to allow rank to grow.Source: §5.6 · p. 32

6 The transverse field Ising model

The section introduces the transverse field Ising (TFI) model as a setting for studying ground states and dynamics, and explains its uses and limits in quantum annealing. It emphasizes that although the model is tractable in several regimes, frustrated instances can make annealing exponentially slow.

6 The transverse field Ising model

  • The TFI model describes N spins coupled by Jij, with fields hZ and hX; the section considers its ground state and dynamics, and hX = 0 gives the classical Ising model.The model is introduced as a genuine quantum many-body problem that naively requires storing a vector of size 2^N for brute-force diagonalization.Source: §6 · p. 33
  • Despite that brute-force cost, 1D nearest-neighbor TFI maps to free fermions, while DMRG gives essentially exact ground states for more complex 1D or quasi-1D interactions.The problem is sign-problem free in all dimensions, enabling quantum Monte Carlo methods.Source: §6 · p. 33
  • Frustrated classical Ising ground-state finding is NP-complete, and discrete optimization problems can be mapped to classical Hamiltonian ground states, motivating quantum annealing by slowly reducing hX.The annealing path starts at high hX, where the ground state is |++···+〉, and aims to follow it down to hX = 0; whether this offers quantum advantage remains under debate.Source: §6 · p. 33
  • For interesting classical spin-glass instances, the minimum ground-to-first-excited-state gap ∆ tends to decrease exponentially with N, making quantum annealing exponentially long.The Landau-Zener transition depends on whether the rate |dhX/dt| is small or large relative to ∆^2.Source: §6 · p. 34

7 Solving Hamiltonian models

The section adapts tensor-network methods to Hamiltonians, showing how to construct an MPO, find its ground state with DMRG, and simulate dynamics with TEBD.

7.1 Direct construction of the MPO

  • For nearest-neighbor couplings, an analytic auto-MPO construction stores pending local-operator information in MPO columns and accumulates the Hamiltonian in the last column.More complex Hamiltonians may require additional remembered terms and higher MPO rank; practical construction may use compression, while TCI is an alternative.Source: §7.1 · pp. 35–36

7.2 DMRG as the diagonalization of an MPO.

  • DMRG diagonalizes an MPO by sweeping over an MPS variational ansatz of moderate bond dimension and locally selecting the lowest eigenvector of the environment matrix.The method optimizes one tensor at a time at the orthogonal center, using upper and lower environments; the approach can diagonalize any MPO, not only many-body Hamiltonians.Source: §7.2 · pp. 36–37

7.3 Quantum dynamics with TEBD

  • At η ≈ 0.1, imaginary-time TEBD converges the N = 30 transverse-field Ising model to its ground state, balancing time and bond dimension.The figure uses hZ = 0, hX = 1, and maximum bond dimension χ = 40; χ = 10 is indistinguishable at the plotted scale. Very small η increases error through accumulated compression steps, while increasing bond dimension can counteract this effect.Source: §7.3 · p. 38, Figure 3 · p. 38

8 MPO and MPS as large matrices and vectors

MPS and MPO provide compressed representations of gigantic vectors and matrices, extending tensor-network methods beyond quantum many-body physics to linear algebra on ultra-large objects.

8 MPO and MPS as large matrices and vectors

  • Sampling, multiplication, lowest-eigenvalue search, and compression form an almost complete toolbox, though success depends on low-rank solution structure.Constructing MPS/MPO representations for general data remains the missing ingredient addressed next by TCI.Source: §8 · pp. 38–39

8.1 Element-wise multiplication between two MPS

  • Element-wise MPS products use a copy tensor and can be approximated by zip-up/DMRG sweeps or TCI; these methods scale as χ^4, while unpublished approaches scale as χ^3.The χ^3 approaches were not public at the time of writing.Source: §8.1 · p. 39

8.2 Adding two MPS

  • Adding two MPS gives a bond dimension equal to the sum of their bond dimensions, so compression may be needed when the sum is redundant.For two identical MPS, the true bond dimension is unchanged.Source: §8.2 · p. 39

8.3 Solving linear problems with MPO/MPS

  • For positive-definite A, an MPO/MPS linear solve minimizes a convex functional tensor by tensor, solving local linear problems and sweeping until convergence.For non-positive-definite A, convergence is not guaranteed; one option is to run the method anyway, and another is to solve A†Ax = A†b, which may increase rank.Source: §8.3 · pp. 39–40

9 Tensor Cross Interpolation for learning tensor networks

Tensor Cross Interpolation (TCI) learns an MPS approximation of a known function by querying selected input configurations rather than accessing the full exponentially large tensor. Its matrix cross-interpolation foundation uses selected rows and columns, and its tensor form can represent functions with only O(Ndχ^2) evaluations.

9 Tensor Cross Interpolation for learning tensor networks

  • TCI maps a known function supplied as a queryable evaluator to an MPS, or an MPO after flattening input and output indices, by actively selecting a tiny fraction of configurations.Unlike gradient-descent neural-network optimization, TCI uses structured tensor representations that support operations such as integration; it is effective only when entanglement is limited.Source: §9 · pp. 40–41

9.1 Another way to factorize matrices

  • Cross interpolation approximates a low-rank matrix using only χ rows and columns, with the Schur complement giving the approximation error and exactness when the matrix has rank χ.The interpolation is exact on the selected blocks; unlike SVD, it does not require access to the full matrix.Source: §9.1 · p. 41, §9.1.2 · pp. 42–43

9.1.3 Practical cross interpolation

  • The pivot rows and columns are chosen iteratively by the maxvol principle, equivalently selecting the pivot with the largest current cross-interpolation error to add the most new information.The approximation is defined through the selected row and column lists.Source: §9.1.3 · p. 43

9.1.4 Stable evaluation of the cross interpolation

  • Partial rank-revealing LU (prrLU) evaluates cross interpolation stably by adding pivots iteratively and stopping after χ pivots, rather than explicitly inverting an increasingly singular pivot matrix.The notes also describe QR factorization as an alternative stabilization method.Source: §9.1.4 · pp. 43–44

9.2 TCI: Extension of CI to n-dimensional tensors

  • Tensor cross interpolation extends matrix cross interpolation by iteratively flattening tensor indices, while the naive factorization is impractical because it requires exponentially large memory and computation.The tensor representation is defined by nested multi-index pivot lists and uses only O(Ndχ^2) entries of the function.Source: §9.2 · p. 45, §9.2.1 · p. 45, §9.2.2 · p. 46

9.2.3 Practical TCI algorithm

  • The practical TCI algorithm initializes a rank-one approximation and sweeps over adjacent tensor pairs, adding pivots until the pivot error falls below a threshold for a full sweep.The initial approximation is exact when the function factorizes into single-variable tensors.Source: §9.2.3 · pp. 46–47

9.2.4 Application to integrals

  • TCI-based multidimensional integration contracts the function MPS with a rank-one weight MPS, reducing the exponentially large quadrature sum to matrix-vector multiplications and, when it works, converging faster than Monte Carlo without its oscillatory-integrand sign problem.For the reported integral, error saturation in some curves reflects insufficient grid density rather than TCI factorization error.Source: §9.2.4 · p. 48
  • For the orange curve, around 8 digits of precision required a few million integrand calls, versus an estimated 1.6 × 10^13 calls for direct integration, a seven-orders-of-magnitude speed-up.The cosine term causes a huge sign problem for Monte Carlo integration.Source: §9.2.4 · p. 48

10 The Quantics representation of functions · 10.1 The basics of quantics · 10.2 Explicit quantics representation of polynomials

The quantics representation encodes function samples as MPS over binary input bits, enabling low-rank function representations and, for suitable problems, an exponentially fine grid at regular-grid cost. The section describes its use in PDE computation with MPS and MPOs.

10 The Quantics representation of functions

  • PDE computations combine quantics MPS discretization, MPO differential operators, TCI-converted inputs, and MPO–MPS plus element-wise multiplication for potential and nonlinear terms.These ingredients are presented for generic PDEs, with the Gross–Pitaevskii equation as a concrete example.Source: §10 · p. 50
  • The PDE dynamics can be solved with MPS and MPO counterparts of methods such as Runge–Kutta, Crank–Nicolson, or spectral approaches.The notes also allow intermediate approaches and replace conventional vectors and matrices with their tensor-network counterparts.Source: §10 · p. 50
  • For suitable problems, quantics yields an exponentially fine grid at the cost of a regular grid.The lectures illustrate this prospect with a Gross–Pitaevskii simulation.Source: §10 · p. 50

10.1 The basics of quantics

  • Quantics writes samples on 2^N equally spaced points as an MPS over the N binary bits of the sample index, whose bits encode different scales.Changing the least-significant bit shifts the index by one, while changing the most-significant bit shifts it by 2^(N−1).Source: §10.1 · pp. 50–51
  • The exponential has rank 1, sine and cosine have rank 2, and a sum of two cosines has rank 4 even when their frequencies differ by orders of magnitude.The function x also has rank 2.Source: §10.1 · p. 51

10.2 Explicit quantics representation of polynomials

  • Any degree-n polynomial has an exact quantics MPS representation of rank n+1, so smooth functions can have low-rank approximate representations through polynomial approximation.The section gives an explicit construction for polynomials on [0,1].Source: §10.1 · p. 51, §10.2 · p. 51
  • Quantics can also represent non-smooth functions efficiently: the delta function has rank 1.Thus low rank is not limited to smooth functions.Source: §10.2 · p. 52

10.3 The magic quantics tensor

The section introduces the “magic quantics tensor,” a local building block for constructing MPOs such as those for finite-difference derivatives and the discrete Laplacian. It encodes binary addition through elementary carry rules, enabling several quantics operations.

10.3 The magic quantics tensor

  • A single “magic quantics tensor” provides a straightforward building block for constructing many quantics MPOs, including finite-difference derivatives and the discrete Laplacian.The section notes that the underlying mathematical constructions appear in.Source: §10.3 · p. 52
  • The tensor is 1 when x = x′, y = x′ + y′ + c′, and c = floor((x′ + y′ + c′)/2), and 0 otherwise; contracting tensors matches successive carry bits.This encodes binary addition, with the output carry at one stage serving as the input carry at the next.Source: §10.3 · p. 53
  • Replacing the initial zero carry with (1,1) changes Θ to perform addition modulo 2^N.The resulting map is Θ_nm,n′m′ = δ_n,n′δ_m,n′+m′[2^N].Source: §10.3 · p. 53
  • Tracing out the n output yields an MPO used to perform convolutions between MPS functions.The resulting network can be contracted using the zip-up algorithm and/or DMRG.Source: §10.3 · pp. 53–54
  • Translations by a vector n = n_N n_{N−1} ··· n_1 are represented with the MPO, and adding the corresponding MPOs for n = 0···001 gives the discrete Laplacian.The section presents these as further uses of the magic quantics construction.Source: §10.3 · p. 54

10.4 Indefinite integral · 10.5 Generalization to higher dimensions

In quantics, indefinite integration is implemented by a rank-2 MPO built from bitwise comparison, while multidimensional representations offer several index orderings whose choice depends on the application.

10.4 Indefinite integral

  • Indefinite integration is as simple as differentiation in quantics: its MPO has rank χ = 2.The lecture contrasts this with conventional integration, which is seldom possible explicitly.Source: §10.4 · p. 54
  • The integral MPO represents Θ(n−m) by comparing corresponding bits from most to least significant and sending a flag once the comparison is decided.Equal bits leave the comparison undecided; the first unequal bits determine the result.Source: §10.4 · pp. 54–55
  • The rightmost tensor is modified to decide the value of μ = Θ(0).Source: §10.4 · p. 55

10.5 Generalization to higher dimensions

  • For multidimensional quantics, indices can be ordered variable by variable or interleaved by grouping indices at the same scale.Source: §10.5 · p. 55
  • Choosing an index ordering depends on the application; in two dimensions, the mirror configuration has often yielded the lowest bond dimension.The mirror configuration combines advantages of the variable-by-variable and interleaved orderings.Source: §10.5 · p. 56
  • Generalizing the mirror configuration to three dimensions requires tree product states and operators, and MPO/MPS algorithms can be extended to trees.The notes describe these tree methods as not particularly difficult and say literature on them is emerging.Source: §10.5 · p. 56

10.6 Application to the Poisson equation · 10.7 Application to the Schrödinger equation

The sections show how quantics tensor networks can solve Poisson and Schrödinger equations by representing inputs and operators as tensor networks. Their efficiency depends on low bond dimensions, while the Schrödinger problem is solved as an MPO eigenvalue problem.

10.6 Application to the Poisson equation

  • For the Poisson equation, choose a quantics representation, use TCI to encode n(r) as an MPS, construct the Laplacian MPO, and solve the resulting linear problem with a tensor-network solver.Source: §10.6 · p. 56
  • When n(r) and the solution U(r) need small bond dimensions, often independent of N, the method reaches an exponentially fine grid at polynomial cost.Whether this advantage holds depends on the bond dimensions required for both the input and solution.Source: §10.6 · p. 56
  • To impose a Dirichlet condition such as a conductor held at constant potential, use the Helmholtz equation and add a diagonal MPO for the density ρ(r) to the Laplacian MPO.The diagonal MPO is obtained by feeding ρ(r) to TCI and using the copy tensor; formally infinite ρ(r) corresponds to a perfect metal.Source: §10.6 · p. 56
  • The boundary-condition method’s efficiency depends on the rank of ρ(r), and even simple geometries such as a sphere may not have low rank in the introduced representations.Finding low-rank quantics representations of geometries remains a difficult problem.Source: §10.6 · p. 57

10.7 Application to the Schrödinger equation

  • The resulting Schrödinger eigenvalue problem is solved with DMRG, replacing the Poisson equation’s linear solver with an eigenvector solver.In this setting, exponential complexity corresponds to an exponentially fine grid.Source: §10.7 · p. 57
  • For the Schrödinger equation, the potential U(r) can come from TCI or a preceding Poisson solution, and the ingredients are represented as MPOs.Source: §10.7 · p. 57

10.8 The quantum Fourier transform as a low-rank MPO

The discrete Fourier transform is a low-rank MPO, and the section gives an explicit QFT-MPO construction and illustrates quantics Fourier methods on the heat equation.

10.8 The quantum Fourier transform as a low-rank MPO

  • The discrete Fourier transform is the quantum Fourier transform and has MPO rank about χ = 15 at machine precision, with χ = 10 typically sufficient in practice.The low-rank MPO has implications for entanglement generation in the corresponding part of a quantum algorithm.Source: §10.8 · p. 57

10.8.1 Explicit construction of the Quantum Fourier Transform MPO

  • The explicit QFT-MPO construction reverses time/frequency bit ordering; approximating the smooth cross-bit phase on [0,1] by polynomials yields χ factorized terms.The construction uses interpolants, with Chebyshev grid points offered as a choice for rapid convergence.Source: §10.8.1 · pp. 58–59

10.8.2 Application to the heat equation

  • For the one-dimensional heat equation, Fourier space gives û(k,t) = e^-k^2t û0(k); TCI encodes the initial condition and kernel before Fourier transformation and diagonal-MPO multiplication yield the solution.The passage notes that the approach is not specific to one dimension.Source: §10.8.2 · pp. 59–60
  • Quantics heat-equation calculations run almost instantaneously on a laptop for grids of up to a billion points; Figure 8 illustrates the computed solution.The scheme uses two TCI calls and three MPO–MPS multiplications.Source: Figure 8 · p. 60, §10.8.2 · p. 60

11 Conclusion

The conclusion emphasizes tensor networks’ versatility while cautioning that other structures and approaches may be better suited to some exponentially large problems. It presents TCI as a bet on new applications beyond many-body physics and potentially on mapping inputs for useful future quantum computers.

11 Conclusion

  • Tensor networks are versatile and powerful, but other structures and approaches can succeed where tensor networks fail on exponentially large problems.The notes stress that the needed structure is not necessarily related to entanglement.Source: §11 · pp. 60–61
  • The notes present MPS/MPO algorithms independently of many-body physics, while leaving out specialized DMRG variants, symmetry adaptations, and tensor-network types beyond MPS and MPO.The authors note that many-body problems have additional structure that algorithms can exploit.Source: §11 · p. 61
  • The authors identify TCI as a bet on emerging non-many-body applications such as quantics for PDEs and on mapping useful future quantum-computer inputs into tensor networks for circuit construction.They describe TCI as enabling problems to be mapped into the tensor-network formalism.Source: §11 · p. 61

A Code repositories

Two code repositories accompany the lectures, providing pedagogical examples and participant-contributed code.

A Code repositories

  • The tutorial repository provides Python examples for hands-on sessions, including elementary gates, full-state and MPS-based quantum computer emulators, and MPS routines.The repository is at https://gricad-gitlab.univ-grenoble-alpes.fr/theorypheliqs/whocancompete-tutorial.Source: A Code repositories · p. 61
  • Chen-How Huang contributed code that generates data for Figs. 1 and 3.The repository is at https://gricad-gitlab.univ-grenoble-alpes.fr/theorypheliqs/whocancompete-example.Source: A Code repositories · pp. 61–62
  • The contributed code covers a GHZ-state timing comparison and an imaginary-time TEBD calculation for the transverse-field Ising model.Source: A Code repositories · p. 62

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