Barren Plateaus in Quantum Linear Combinations: Interference and Lie-Algebraic Control

Amir Koutahi; Pouya Kananian; Hans-Arno Jacobsen

quant-phOct 28, 2026

Barren Plateaus in Quantum Linear Combinations: Interference and Lie-Algebraic Control

Authors: Amir Koutahi, Pouya Kananian, Hans-Arno Jacobsen

Organizations: Division of Engineering Science, University of Toronto · Department of Computer and Mathematical Sciences, University of Toronto Scarborough · Department of Electrical and Computer Engineering, University of Toronto

Abstract

A coherent sum of quantum circuits introduces interference that can alter the training signal. We distinguish a single linear combination of unitaries (LCU) from a circuit that repeatedly alternates branch operations and ancilla mixing. For a single-pass LCU, independent branches with zero matrix means cannot erase the differentiated branch's gradient variance: the total variance is at least the branch variance times the fourth power of its coefficient magnitude. This bound holds at any gate position and does not require the circuit unitaries to be Haar-random. Beyond this lower bound, we derive exact loss and gradient formulas that quantify the additional contributions from interference between branches. One of these terms depends on relative coefficient phases and vanishes when the branch ensemble is invariant under multiplication by $i \mathbb{I}$ ; a four-qubit matchgate example shows that it can be nonzero. For repeated interleaving with full special-unitary ancilla control, we determine the generated Lie algebra from the associative algebra of the branches, including the available central phase directions. This control result predicts concentration only with additional assumptions on circuit sampling. Matchgate examples exhibit polynomial single-pass LCU variances and exponential concentration under joint Haar-random sampling. We also derive coefficient-selection rules and bounds for postselection, measurement error, and approximate moments.

Figures & tables

Panel of Figure 1 — Figure 1: Two circuit models. (a) A single-pass LCU prepares an ancilla superposition (PREP), applies each branch unitary conditioned on its label (SELECT), and unprepares the ancilla. Projection onto the reference ancilla state gives the block $A/\lambda$ . Its fluctuations are determined by branch and interference moments. (b) Repeated interleaving applies controlled branch layers $S_{t}$ and ancilla mixers $C_{t}$ in alternation; its projected block generally contains products of branch operations. The available generators determine reachable control. A variance prediction additionally requires the actual ensemble moments. The detailed definitions appear in Section 3.6 .

Symbol	Lives in	Definition	Role
$\mathfrak{g}_{\ell}$	$\mathfrak{u}(d)$	$\operatorname{Lie}_{\mathbb{R}}\{iH_{\ell,r}\}_{r}$	DLA of branch $\ell$ (branch layer)
$\mathfrak{g}_{\mathrm{coeff}}$	$\mathfrak{u}(L)$	$\operatorname{Lie}_{\mathbb{R}}\{iK_{\xi}\}_{\xi}$ ; equal to $\{0\}$ for fixed coefficients	DLA of the coefficient layer, acting on the ancilla alone
$\Delta_{a}$	$\mathfrak{u}(L)$	$\operatorname{span}_{\mathbb{R}}\{iP_{\ell}\}_{\ell}$	diagonal ancilla generators: no mixing between branches
$\mathfrak{c}$	$\mathfrak{u}(Ld)$	$\mathfrak{g}_{\mathrm{coeff}}\otimes\mathbb{I}_{s}$	coefficient layer embedded in the joint space
$\mathfrak{h}$	$\mathfrak{u}(Ld)$	$\bigoplus_{\ell}P_{\ell}\otimes\mathfrak{g}_{\ell}$	branch layer embedded in the joint space; block diagonal in the ancilla index
$\mathfrak{g}_{\mathrm{abs}}$	$\mathfrak{g}_{\mathrm{coeff}}$	$\{K\in\mathfrak{g}_{\mathrm{coeff}}:K\otimes\mathbb{I}_{s}\in\mathfrak{h}\}$	coefficient directions already supplied by the branch layer

Table 1: Top: the algebras of Section 5 . Bottom: their dimensions for two examples with

L=2

branches on

n=1

qubit (

d=2

), for four coefficient layers each; “lower” and “upper” are the lower and upper dimension bounds,

\sum_{\ell}\dim\mathfrak{g}_{\ell}

and

\dim_{\mathbb{C}}\mathcal{A}_{a}\cdot\dim_{\mathbb{C}}\mathcal{A}_{s}

. Example A has

\mathfrak{g}_{1}=\mathfrak{g}_{2}=\operatorname{span}_{\mathbb{R}}\{iZ\}

; Example B has

\mathfrak{g}_{1}=\operatorname{span}_{\mathbb{R}}\{iX\}

and

\mathfrak{g}_{2}=\operatorname{span}_{\mathbb{R}}\{iZ\}

, with

X,Z

the single-qubit Pauli matrices. Both have

\dim\mathfrak{h}=2

. In Example A,

\mathcal{A}_{s}

has

q=2

one-dimensional blocks; in Example B,

\mathcal{A}_{s}=M_{2}(\mathbb{C})

is a single block with

d_{1}=2

. In Example A the branches share an abelian algebra, so a mixer can only reach

\mathfrak{su}(2)\oplus\mathfrak{su}(2)

plus a center; in Example B the branches are complementary and the mixer saturates

\mathfrak{su}(4)

, respectively

\mathfrak{u}(4)

, even though the joint Lie closure of the two branches is only

\mathfrak{su}(2)

(Example 5.5 ). A diagonal layer adds exactly the central directions counted by ( 46 ), and

\mathfrak{g}_{\mathrm{abs}}=\{0\}

throughout because no branch algebra contains

i\mathbb{I}_{s}

Panel of Figure 2 — Figure 2: Loss and gradient benchmarks for $L=2$ , $n\geq 2$ , $\rho=|0\rangle\langle 0|^{\otimes n}$ , and $O=Z_{1}$ . (a) Single-pass LCU loss with independent full-matchgate Haar-random branches: balanced coefficients give $\operatorname{Var}(\mathcal{L})/\lambda^{4}=\tfrac{1}{8}[(2n-1)^{-1}+2^{1-n}]$ ; matched Haar-random preparation gives uniform simplex weights and $\operatorname{Var}(\mathcal{L})=\tfrac{2}{5}(2n-1)^{-1}+\tfrac{1}{15}2^{1-n}$ at $\lambda=1$ , recovering the $L=2$ specialization of Eq. (17) in Khatri et al. [ 6 ] . (b) Balanced single-pass LCU gradient for $U_{1}=e^{i\theta X_{1}X_{2}}V_{1}$ : the exact variance and its branch floor are evaluated in ( 57 ). (c) Joint Haar-random loss of the interleaved model: parity-block and universal values are $[2(2^{n}+1)]^{-1}$ and $[2(2^{n+1}+1)]^{-1}$ after dividing by $\lambda^{4}$ . The universal curve is a separate benchmark for branches with a full joint system envelope; matchgate branches retain parity and cannot reach that algebra. The vertical ranges differ; the vertical axes are logarithmic. The horizontal axis is qubit count, not depth.

Argument	Hypotheses to check	Proof location
Gradient floor, Proposition 4.1	Fixed coefficients; independent branch data; zero means of the other branches; finite second moments	Section 4.1
Single-pass LCU loss and pairings	Independence and zero means for the decomposition; required Haar-measure moments for invariant pairings	Appendix B
Single-pass LCU and joint gradients	Stated gate placement and ensemble factorization for closed forms; membership for sector formulas	Appendix C
Bounds, saturation, graph control	Specified generator families; full special-unitary mixer for saturation; central directions for the exact upper bound	Appendix D
Joint-space variance	Joint first and second moments; state or observable membership for the sector formula	Appendix E
Matchgate and Pauli examples	Actual branch groups and envelopes; specified input, observable, and ensemble	Appendix F

Table 2: Assumptions and proof locations. Structural results require no ensemble law. A large algebra alone is not a concentration hypothesis. Statements in the main text also link directly to their proofs.

Barren Plateaus

Barren plateaus are quantum-circuit optimization landscapes where gradients vanish with system size, hindering the training of variational algorithms.

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Machine Learning Research Group, University of Oxford, United Kingdom · 2Quantinuum, London, United Kingdom

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Variational quantum circuits have been central to many proposed near-term applications of quantum computing, but a growing body of evidence suggests that trainability and quantum advantage are fundamentally at odds: ansätze expressive enough to resist efficient classical simulation tend to exhibit barren plateaus, while structures that provably rule out barren plateaus typically render them classically simulable. We propose a stacked linear combination of unitaries (S-LCU) as a variational ansatz which provides a tunable trade-off between barren plateaus and classical simulability. Using a diagrammatic analysis, we bound the loss-landscape variance of the Free Fermion S-LCU, whose elements are fermionic Gaussian unitaries. We prove a variance lower bound of

Ω(1/(n k^{3l}))

, with a simulation cost of

O(k^{2l} n^3)

using the best known classical algorithm, compared to a quantum gate complexity of only

O(lkn^2)

. The number of layers

l

serves as a single dial that trades computational complexity against the rate of cost concentration. This offers practitioners a systematic method for constructing ansätze with a complexity-trainability trade-off that best suits their application and hardware.

Abstract

Barren Plateaus

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