We prove that variational quantum algorithms using discrete variable boson samplers suffer from barren plateaus and that by changing to fermionic statistics, which is classically efficient to simulate, the barren plateau scaling is less severe. Moreover, when using fermionic statistics the cost landscape boils down to one unit frequency Fourier mode per variational parameter, irrespective of circuit design, ansatz or variational problem, allowing for more efficient parameter shift rules. This simplified cost landscape allows for the gradient-free Rotosolve algorithm to be used to gain a considerable speed-up over the original bosonic algorithm. Motivated by this, we propose a non-linear parametrised phase shifter, the dual-valued phase shifter, that generates these single-harmonic cost landscapes for bosonic statistics. We propose three ways to achieve this by using either non-linear optics, measurement-induced non-linearities, or entangled resource states simulating fermionic statistics. The latter two require linear optics only, allowing for implementation with widely-available technology today. Despite this, the classical simulability of free fermions and the fact the fermionic algorithm can defeat the bosonic algorithm casts doubt on whether the quantum advantage of boson sampling results in any practical advantage in this setting.
We prove that variational quantum algorithms using discrete variable boson samplers suffer from barren plateaus and that by changing to fermionic statistics, which is classically efficient to simulate, the barren plateau scaling is less severe. Moreover, when using fermionic statistics the cost landscape boils down to one unit frequency Fourier mode per variational parameter, irrespective of circuit design, ansatz or variational problem, allowing for more efficient parameter shift rules. This simplified cost landscape allows for the gradient-free Rotosolve algorithm to be used to gain a considerable speed-up over the original bosonic algorithm. Motivated by this, we propose a non-linear parametrised phase shifter, the dual-valued phase shifter, that generates these single-harmonic cost landscapes for bosonic statistics. We propose three ways to achieve this by using either non-linear optics, measurement-induced non-linearities, or entangled resource states simulating fermionic statistics. The latter two require linear optics only, allowing for implementation with widely-available technology today. Despite this, the classical simulability of free fermions and the fact the fermionic algorithm can defeat the bosonic algorithm casts doubt on whether the quantum advantage of boson sampling results in any practical advantage in this setting.
Python software and data for the publication "Mitigating the barren plateau problem in linear optics" by Matthew D. Horner. To run this Python code, first install Aegiq's Lightworks and the fermion sampling extension. pip install lightworks pip install lightworks[fermionic] Notebooks This contains notebooks for various figures. Run each notebook to generate the data. Figure 2 - Barren plateau scaling of bosonic and fermionic linear optics. Figure 5 - Measurement-induced dual-valued phase shifter (DVPS). Figure 8 - False herald probasbility calculation for the DVPS Figure 9 - Cost landscape comparison. Figure 10 - Solving a random QUBO problem using gradient descent or Rotosolve for either bosonic or fermionic statistics. Figure 12 - Calculation of the Fourier modes in the QUBO cost landscape. Raw data This also contains raw data for Figure 2 - The barren plateau scaling figures stored in three .npz files Figure 5 - A set of pre-computed DVPS unitaries in a single .npz file Figure 9 - The raw cost landscape data in a single .npz file. Figure 12 - All subfigure data in 12 .txt files. This data is produced by the notebooks so is not necessary, but is included to save time. Scripts The scripts lightworks_optimiser_classes.py lightworks_qubo.py contain code for optimising via gradient descent and Rotosolve. These are called by the notebooks.
Variational quantum algorithms, which combine highly expressive parameterized quantum circuits (PQCs) and optimization techniques in machine learning, are one of the most promising applications of a near-term quantum computer. Despite their huge potential, the utility of variational quantum algorithms beyond tens of qubits is still questioned. One of the central problems is the trainability of PQCs. The cost function landscape of a randomly initialized PQC is often too flat, asking for an exponential amount of quantum resources to find a solution. This problem, dubbed barren plateaus , has gained lots of attention recently, but a general solution is still not available. In this paper, we solve this problem for the Hamiltonian variational ansatz (HVA), which is widely studied for solving quantum many-body problems. After showing that a circuit described by a time-evolution operator generated by a local Hamiltonian does not have exponentially small gradients, we derive parameter conditions for which the HVA is well approximated by such an operator. Based on this result, we propose an initialization scheme for the variational quantum algorithms and a parameter-constrained ansatz free from barren plateaus.
We investigate the barren plateau (BP) phenomenon in variational quantum algorithms using a statistical approach. Using Gaussian function models, we identify three distinct types of BPs. The first type, which we called localized-dip BPs, occurs in landscapes that are mostly flat but contain a dip point where the gradient is large in a small region around the minimum. The second type, called localized-gorge BPs, which are somewhat similar to the localized-dip BPs but contain a gorge line. The third type, called everywhere-flat BPs, appears when the entire landscape is uniformly flat with almost vanishing gradients, making optimization significantly more difficult. After illustrating these behaviors in the Gaussian function models, we extend the analysis to the variational quantum eigensolver (VQE). We consider two types of ansätze: the hardware-efficient ansatz and the random Pauli ansatz. For both ansätze, we only observe the everywhere-flat BPs. Using our statistical approach, we searched for localized-dip and localized-gorge BPs but found no evidence of such features in the examples studied, suggesting that everywhere-flat BPs dominate in these ansätze. Our method effectively probes landscape features by capturing the gradient scaling across parameter space, making it a useful tool for diagnosing BPs in variational algorithms. To mitigate BPs in the VQE, we employ a genetic algorithm (GA) to optimize the random gates generated in the ansätze, thereby reshaping the cost function landscape to enhance the optimization efficiency. A comparison with an unoptimized ansatz shows how the ansatz design can improve the scalability and reliability of variational quantum algorithms.
A large amount of effort has recently been put into understanding the barren plateau phenomenon. In this perspective article, we face the increasingly loud elephant in the room and ask a question that has been hinted at by many but not explicitly addressed: Can the structure that allows one to avoid barren plateaus also be leveraged to efficiently simulate the loss classically? We collect evidence-on a case-by-case basis-that many commonly used models whose loss landscapes avoid barren plateaus can also admit classical simulation, provided that one can collect some classical data from quantum devices during an initial data acquisition phase. This follows from the observation that barren plateaus result from a curse of dimensionality, and that current approaches for solving them end up encoding the problem into some small, classically simulable, subspaces. Thus, while stressing that quantum computers can be essential for collecting data, our analysis sheds doubt on the information processing capabilities of many parametrized quantum circuits with provably barren plateau-free landscapes. We end by discussing the (many) caveats in our arguments including the limitations of average case arguments, the role of smart initializations, models that fall outside our assumptions, the potential for provably superpolynomial advantages and the possibility that, once larger devices become available, parametrized quantum circuits could heuristically outperform our analytic expectations. There has been growing evidence that strategies to circumvent the barren plateau problem in variational quantum computing might also kill potential quantum advantages. In this Perspective, the authors gather this evidence, discuss what is still missing to provide a definitive answer, and provide new research directions.
Variational quantum circuits have recently gained much interest due to their relevance in real-world applications, such as combinatorial optimizations, quantum simulations, and modeling a probability distribution. Despite their huge potential, the practical usefulness of those circuits beyond tens of qubits is largely questioned. One of the major problems is the so-called barren plateaus phenomenon. Quantum circuits with a random structure often have a flat cost-function landscape and thus cannot be trained efficiently. In this paper, we propose two novel parameter conditions in which the hardware-efficient ansatz (HEA) is free from barren plateaus for arbitrary circuit depths. In the first condition, the HEA approximates to a time-evolution operator generated by a local Hamiltonian. Utilizing a recent result by [Park and Killoran, Quantum 8, 1239 (2024)], we prove a constant lower bound of gradient magnitudes in any depth both for local and global observables. On the other hand, the HEA is within the many-body localized (MBL) phase in the second parameter condition. We argue that the HEA in this phase has a large gradient component for a local observable using a phenomenological model for the MBL system. By initializing the parameters of the HEA using these conditions, we show that our findings offer better overall performance in solving many-body Hamiltonians. Our results indicate that barren plateaus are not an issue when initial parameters are smartly chosen, and other factors, such as local minima or the expressivity of the circuit, are more crucial.
Whether a fermionic variational quantum algorithm offers a genuine quantum advantage depends on whether its outputs are classically hard to compute. We show that for such algorithms classical simulability is not an intrinsic property of the circuit. It depends on the fermion-to-qubit encoding. Standard encodings are related by Clifford transformations, so they form a gauge group, and we classify the standard simulability resources by how they transform under it. The Pauli weight, the causal-cone width, and the entanglement that sets the cost of tensor-network contraction are gauge-covariant: a generic interacting Hamiltonian whose truncated Pauli-propagation cost grows linearly in the system size under the Jordan-Wigner encoding becomes logarithmic under a tree encoding, and we exhibit the resulting change of verdict at a fixed physical ansatz. The dynamical-Lie-algebra dimension and the non-stabilizerness are gauge-invariant. From this split we prove that the encoding-optimized simulation cost isbounded by the invariant resources, so genuine, encoding-robust hardness requires both the dynamical Lie algebra and the magic to be large; the covariant weight is necessary but never sufficient. These routes are sufficient certificates of simulability, not a complete characterization, so the floor is an upper bound and a full classification of representation-invariant cost functionals is left open. We confirm both invariant simulation routes numerically, and we show that the dynamical-Lie-algebra dimension separates simulable from hard instances where the Pauli weight cannot; we also construct an explicit one-parameter interacting family for which all three certificate routes fail at once, so the floor is achieved. We then establish a sharp dichotomy with trainability. Because the cost landscape is encoding-invariant, the gradient and the barren-plateau variance are exactly gauge-invariant, which we verify to machine precision; we show further that this invariance follows from faithfulness rather than Clifford-equivalence and is robust even to symmetry tapering, so trainability can be altered only by changing what the dynamics explores, such as restricting the ansatz, never by relabeling the qubits. One may therefore re-encode a fermionic circuit to make it easier to simulate, but never to make it easier to train. These results reframe advantage claims for fermionic ansatze: an advantage that isnot gauge-invariant is an artifact of the encoding