Sequential Quantum Circuits Overview

• Sequential quantum circuits are defined by local operations applied in a sequence, allowing efficient state preparation and controlled entanglement across quantum systems.
• They enable advanced simulation techniques, topological code-state preparations, and phase mappings by leveraging sequential locality in circuit constructions.
• These circuits offer flexibility in algorithm design while facing challenges such as noise-induced barren plateaus and error accumulation in deep quantum circuits.

Sequential quantum circuits (SQC) are circuit constructions in which quantum operations are organized sequentially across a system or across iterations of a computation, rather than only as finite-depth parallel layers. In contemporary literature, the term covers several closely related but non-identical ideas: local-unitary circuits that act on subregions in sequence, 1D nearest-neighbor layouts for matrix-product-state preparation and time evolution, while-loop-induced feedback circuits with retained internal quantum state, and application-specific sequential block constructions for simulation and code-state preparation (Chen et al., 2023, Wei et al., 18 Mar 2025, Astrakhantsev et al., 2022, Li et al., 14 May 2026, Hu et al., 7 Mar 2025).

1. Terminology and scope

Current usage is plural rather than singular. In one condensed-matter definition, a quantum circuit is called an SQC if each local degree of freedom is only acted upon by a finite number of gates in the whole circuit. In 1D variational and state-preparation work, an $M$-layer sequential circuit is a repeated arrangement of neighboring gates shifted along a chain. In program-semantics work, an SQC is the circuit induced by a while-loop, with explicit internal and external qubits. Two nearby labels are distinct: "Sequential Quantum Computing" denotes hybrid workflows across multiple processors, whereas "Quantum Sequential Circuits" denotes a hardware-oriented paradigm based on quantum transistors, Choi states, and ebits.

Label in the literature	Characteristic usage	Representative source
Sequential Quantum Circuits	Each local degree of freedom is only acted upon by a finite number of gates in the whole circuit	(Chen et al., 2023)
Sequential circuits in 1D/MPS settings	Repeatedly stacking the same arrangement of gates on neighboring pairs	(Wei et al., 18 Mar 2025)
Sequential circuit semantics for while-loops	Explicit internal and external qubits with feedback/state retention	(Li et al., 14 May 2026)
Sequential Quantum Computing	Multiple homogeneous or heterogeneous quantum processors in hybrid classical-quantum workflows	(Romero et al., 25 Jun 2025)
Quantum Sequential Circuits	Hardware-oriented paradigm built upon the quantum transistor	(Wang, 5 Feb 2026)

The acronym SQC is also used for "Superconducting Quantum Circuits" and for "single-qubit classifiers"; these are separate usages (Pandey et al., 2023, Pasini et al., 2 Jul 2026).

2. Formal circuit models and architectures

A defining formal property of the many-body SQC is sequential locality: local patches, strips, or other sub-regions are acted upon one after another, while every local degree of freedom is only acted upon by a finite number of gates in the whole circuit. This enlarges the circuit class beyond finite-depth quantum circuits; the relation summarized in the literature is

$$\mathrm{FDQC} \subset \mathrm{QCA} \subset \mathrm{SQC} \subset \mathrm{LDQC}.$$

Unlike FDQC and QCA, generic SQCs are not locality-preserving: local operators can be mapped to highly nonlocal ones (Chen et al., 2023).

In a 1D qubit chain, an $M$-layer sequential circuit consists of repeatedly stacking the same arrangement of gates on neighboring pairs, each block sequentially shifted along the chain, so that

$$|\Psi_{\rm SEQ}(\bm{\theta},N,M)\rangle = U_{\rm SEQ}(\bm{\theta},N,M)\,|0\rangle^{\otimes N}.$$

For $M=1$, the single-layer sequential circuit on qubits exactly produces the set of matrix product states with bond dimension $D=2$ (Wei et al., 18 Mar 2025).

A second 1D formulation is the dense or layered uniform sequential circuit. The dense USC is equivalent to a uniform MPS with bond dimension $\chi = 2^{N_q-1}$, while the layered USC decomposes the local unitary into $M_U$ layers of sequential two-qubit gates acting on neighboring pairs. This layered form uses at most $15(N_q-1)M_U$ parameters, compared to $2^{2N_q+1}$ for a dense unitary (Astrakhantsev et al., 2022).

Recent MPS-preparation pipelines combine heuristic staircase-like and brick wall disentangler circuits within a single framework. In that setting, optimized brick wall circuits typically achieve the lowest depths, while the optimized staircase-like circuits minimize gate counts, and low-level optimizations reduce depths by up to 50% and CNOT counts by 33% (Szołdra et al., 12 Feb 2026).

3. Expressivity, entanglement, and noise

Sequentiality has a characteristic entanglement profile. SQCs preserve the entanglement area law and hence the gapped-ness of quantum states, yet the linear-depth structure can change long-range correlation and entanglement and can map between distinct phases (Chen et al., 2023).

For 1D sequential circuits, the entanglement entropy across any cut is bounded as

$$\mathrm{FDQC} \subset \mathrm{QCA} \subset \mathrm{SQC} \subset \mathrm{LDQC}.$$0

At the same time, correlations in an $$\mathrm{FDQC} \subset \mathrm{QCA} \subset \mathrm{SQC} \subset \mathrm{LDQC}.$$1-layer SQC can extend across the whole system. This expressive power interacts sharply with noise: because sequential circuits have depth $$\mathrm{FDQC} \subset \mathrm{QCA} \subset \mathrm{SQC} \subset \mathrm{LDQC}.$$2, idling errors accumulate, the error density satisfies $$\mathrm{FDQC} \subset \mathrm{QCA} \subset \mathrm{SQC} \subset \mathrm{LDQC}.$$3, and the gradient variance obeys

$$\mathrm{FDQC} \subset \mathrm{QCA} \subset \mathrm{SQC} \subset \mathrm{LDQC}.$$4

yielding noise-induced barren plateaus on large devices (Wei et al., 18 Mar 2025).

Uniform sequential circuits display a different resource profile when used as variational ansätze for infinite translation-invariant dynamics. For a given accuracy, the number of parameters required by the l-USC ansatz grows polynomially in the simulation time; the state circuit scales linearly in the reachable time and the environment circuit scales quadratically. All required overlaps, observables, and derivatives are expressible via finite circuits involving only $$\mathrm{FDQC} \subset \mathrm{QCA} \subset \mathrm{SQC} \subset \mathrm{LDQC}.$$5 qubits (Astrakhantsev et al., 2022).

4. Phase mappings, dualities, and topological order

One of the distinguishing uses of SQCs is as maps between gapped phases. Explicit constructions were given for mappings from product states to GHZ states, symmetry-protected topological states, intrinsic topological states, and fracton states. The physical interpretation is tied to condensation, Kramers-Wannier duality, and foliation for fracton phases (Chen et al., 2023).

In the study of quantum many-body scarring, an SQC implementing Kramers-Wannier duality is used to track the fate of low-entanglement excited states under a non-invertible symmetry. The symmetric scar $$\mathrm{FDQC} \subset \mathrm{QCA} \subset \mathrm{SQC} \subset \mathrm{LDQC}.$$6 is mapped to $$\mathrm{FDQC} \subset \mathrm{QCA} \subset \mathrm{SQC} \subset \mathrm{LDQC}.$$7 and back, whereas the antisymmetric scar $$\mathrm{FDQC} \subset \mathrm{QCA} \subset \mathrm{SQC} \subset \mathrm{LDQC}.$$8 is mapped to a thermal state in a sector where no scars are present. The same work argues that non-invertible dualities provide both a generative mechanism for new QMBS and a diagnostic for their stability (Fontana et al., 7 Aug 2025).

A broader invariant-based formulation appears in the theory of sequential unitary circuits for moving non-invertible defects. There, a Berry phase computed from a sequence of defect-moving circuits is invariant under allowed local deformations provided the sequential circuits preserve the locality of those deformations, and a nontrivial invariant rules out a short-range-entangled state that preserves the non-invertible symmetry, thereby signaling an 't Hooft anomaly. Applied to loop excitations in the $$\mathrm{FDQC} \subset \mathrm{QCA} \subset \mathrm{SQC} \subset \mathrm{LDQC}.$$9D $M$0 topological order, a 24-step sequence yields $M$1 for a non-Abelian fermionic loop and $M$2 for the bosonic loop (Sato et al., 10 Jun 2026).

Sequential circuits are also used directly for topological code-state preparation. The Seed-Entangler-Enriched Sequential Quantum Circuit prepares code states of Tetra-Digit models, a family that encompasses Toric Codes across arbitrary dimensions and subsumes the X-cube fracton code as special cases. Its key ingredient is the seed-entangler acting on seeds, the circuit depth scales linearly in the linear system size, and for the constructed examples the number of available seeds equals the number of logical qubits (Hu et al., 7 Mar 2025).

5. Algorithms, programming semantics, and verification

Software and verification work recasts SQCs as objects with explicit feedback and state retention. QSeqSim assigns while-loops a precise sequential circuit semantics with explicit internal and external qubits, translates Qiskit QuantumCircuit objects into OpenQASM 3, and organizes the resulting program into a combination of combinational, dynamic, and sequential circuits. Its symbolic backend uses Binary Decision Diagrams and weighted model counting, and in the quantum random walk benchmark it successfully simulates circuits with over 1000 qubits for more than 10 loop iterations (Li et al., 14 May 2026).

Formal verification has been developed through quantum state machines, a quantum generalisation of Mealy machines. The main theorem on equivalence checking shows that two quantum Mealy machines need only be tested on input sequences taken from a chosen basis and of length quadratic in the dimensions of their state Hilbert spaces. The resulting algorithm for sequential quantum circuits has running time

$M$3

where $M$4 and $M$5 denote the numbers of input and internal qubits (Wang et al., 2018).

6. Specialized realizations and neighboring paradigms

Sequential block constructions also appear in application-driven quantum algorithm design. For quantum computational fluid dynamics, a four-qubit surrogate quantum circuit was trained to approximate the full Bhatnagar Gross Krook collision operator for the D2Q9 lattice while respecting mass and momentum conservation, D8 equivariance and scale equivariance. The 15 block implementation compiles to 2,430 native gates on the IBM Heron gate set under full qubit connectivity, uses neither ancilla qubits nor post-selection or repeated executions, and has depth independent of the grid resolution because collision is a local operation (Lăcătuş et al., 16 Jul 2025).

A distinct but nearby paradigm is Sequential Quantum Computing, which uses multiple homogeneous or heterogeneous quantum processors in hybrid classical-quantum workflows. Information is transferred through bias fields, so that measurement outcomes from one processor are encoded in the initial-state preparation of the subsequent quantum computer. In the reported experiment, a D-Wave quantum annealer using 678 qubits approximately solved a higher-order Ising problem and an IBM 156-qubit digital quantum processor refined the solutions; the hybrid protocol reached Best AR 100% with Best Energy -186.86 using 8,000 shots (Romero et al., 25 Jun 2025).

Another distinct proposal is Quantum Sequential Circuits, a hardware-oriented paradigm built upon the quantum transistor. In that framework, gates are encoded as Choi states via channel-state duality, activated through bulk measurements, and ebits implement the functional analog of feedback loops and memory. The proposal is presented as complementing the existing combinational quantum circuit model and as a route toward a quantum von Neumann architecture (Wang, 5 Feb 2026).