Adaptive Monitored Quantum Circuits

• The paper demonstrates that integrating mid-circuit measurements with classical feedforward enables active control of non-unitary dynamics and entanglement transitions.
• It shows that adaptive circuits can prepare complex many-body states in constant depth, outperforming traditional unitary circuits in tasks like toric-code state preparation.
• Experimental and simulation results validate noise resilience and critical scaling, confirming the practical advantages of adaptive feedback on NISQ hardware.

Searching arXiv for papers on adaptive monitored quantum circuits and closely related monitored/adaptive circuit topics. {"query":"adaptive monitored quantum circuits measurements feedback entanglement phase transitions site:arxiv.org", "max_results": 10} I found relevant recent arXiv papers matching the topic, including experimental and theoretical work on adaptive monitored circuits, state-complexity in adaptive circuits, constant-depth MPS preparation with adaptive circuits, postselection-free monitored dynamics, and noise resilience in adaptive monitored quantum circuits. Adaptive monitored quantum circuits are quantum circuits in which local unitary evolution is interleaved with mid-circuit measurements and classical feedforward, so that later operations depend on earlier measurement outcomes. In the formulations used across recent work, they realize explicitly non-unitary dynamics, can reset or discard measured degrees of freedom, and can steer trajectories toward designated target states or phases. They have been studied as a framework for controlling quantum chaotic dynamics, preparing many-body entangled states in constant depth, characterizing state complexity and quantum phases, accessing monitored-circuit entanglement transitions without postselection, and probing the robustness of symmetry-protected dynamical phenomena on NISQ hardware (Pokharel et al., 22 Sep 2025).

1. Formal definitions and circuit primitives

A common formal model specifies an $n$-qubit target register together with $A$ ancillas, so that the total system has $m=n+A$ qubits. A depth-$L$ adaptive circuit consists of $L$ layers of disjoint $K$-fan-in unitary gates or single-qubit measurements in the computational basis; measured qubits are immediately discarded, and the classical outcomes $s\in\{0,1\}^A$ may be used to classically control subsequent layers. In this description, starting from $|0\rangle^{\otimes m}$, the final post-selected $n$-qubit state is written as

$$|\psi\rangle \propto (I_n\otimes \langle s|)\,U_L\cdots U_2 U_1|0^m\rangle,$$

with depth $A$0 and ancilla count $A$1 as the main resource parameters (Liu et al., 21 Sep 2025).

A trajectory-based formulation emphasizes sequential measurement-conditioned evolution. In a depth-$A$2 adaptive monitored circuit, round $A$3 first applies an outcome-dependent unitary $A$4, then measures a subset of qubits by a POVM $A$5, records outcome $A$6, and allows the next-round unitary $A$7 to depend on it. This makes the circuit depth a property of the arrangement of the two-qubit gates in the unitary layers, while classical processing and feed-forward are external to that depth count (Foss-Feig et al., 2023).

Concrete architectures instantiate these primitives differently. In the experimentally realized adaptive Bernoulli-map circuit, the system is a one-dimensional open chain of $A$8 qubits in discrete time; at each step $A$9, one chooses a site $m=n+A$0 according to a biased random walk and applies either a two-qubit scrambler $m=n+A$1 with probability $m=n+A$2 or a one-qubit control operation $m=n+A$3 with probability $m=n+A$4. The control operation performs a mid-circuit $m=n+A$5-measurement on qubit $m=n+A$6, records $m=n+A$7, and conditionally applies $m=n+A$8 if $m=n+A$9, implementing a reset to $L$0 on that qubit. The full wavefunction after $L$1 steps is

$L$2

with each $L$3 equal to $L$4 or $L$5 (Pokharel et al., 22 Sep 2025).

2. Conditional feedback and non-unitary control

The defining operational feature of adaptive monitored circuits is conditional feedback. In the Bernoulli-map implementation, measurement operators on qubit $L$6 are

$L$7

and the feedback unitary is $L$8 for outcome $L$9 and $L$0 for $L$1. Equivalently, the reset channel is

$L$2

implemented by measure-and-conditional-$L$3. In this architecture, the target fixed point is the fully polarized state $L$4, while scrambling tends toward an ETH-like highly entangled “infinite-temperature” wavefunction with $L$5 (Pokharel et al., 22 Sep 2025).

A distinct feedback protocol appears in the postselection-free approach to monitored dynamics in $L$6-symmetric circuits. There, one first fixes a target trajectory by specifying all target measurement outcomes $L$7. During each shot, whenever the actual outcome $L$8 differs from the target value $L$9, one immediately applies a Pauli-$K$0 gate on that qubit, thereby steering the post-measurement state back to the same local outcome as the target. Repeating this $K$1 times builds a mixed ensemble $K$2, which is then projected to the target total-charge sector $K$3 to yield $K$4. The stated overhead for fixed accuracy $K$5 and circuit size $K$6 scales as $K$7, replacing exponential postselection costs with a polynomial overhead (Pöyhönen et al., 2024).

Adaptive feedback also stabilizes absorbing-state dynamics. In the symmetry-protected brickwork architecture, each qubit is measured in the computational basis with probability $K$8, using

$K$9

and the corrective feedback unitaries are

$s\in\{0,1\}^A$0

The target absorbing state is $s\in\{0,1\}^A$1, and the adaptive map is constructed so that feedback restores the locally desired occupation after measurement (Ivaki et al., 2024).

3. Dynamical phases, chaos control, and critical scaling

Adaptive monitored circuits have provided a direct setting for studying dynamical phase transitions generated by competition between scrambling and feedback. In the quantum Bernoulli-map circuit, the scrambler $s\in\{0,1\}^A$2 approximately enacts a random two-bit mixing that, on computational-basis states, implements the analogue of doubling the bit string. Without control, this rapidly scrambles information and generates volume-law entanglement. The competition parameter $s\in\{0,1\}^A$3 sets the average Lyapunov exponent in the classical limit,

$s\in\{0,1\}^A$4

For $s\in\{0,1\}^A$5, $s\in\{0,1\}^A$6 corresponds to chaotic expansion; for $s\in\{0,1\}^A$7, $s\in\{0,1\}^A$8 corresponds to contraction and drift to the fixed point. In the long-time, large-$s\in\{0,1\}^A$9 limit, the fidelity $|0\rangle^{\otimes m}$0 tends to $|0\rangle^{\otimes m}$1 for $|0\rangle^{\otimes m}$2 and to $|0\rangle^{\otimes m}$3 for $|0\rangle^{\otimes m}$4 (Pokharel et al., 22 Sep 2025).

Three regimes were identified as $|0\rangle^{\otimes m}$5 increases: $|0\rangle^{\otimes m}$6, with volume-law entanglement and $|0\rangle^{\otimes m}$7; $|0\rangle^{\otimes m}$8, with area-law entanglement but still $|0\rangle^{\otimes m}$9; and $n$0, the “controlled” phase, where entanglement becomes subextensive and tends to zero while $n$1. The order parameter is the magnetization density

$n$2

and the circuit-dependent shot-to-shot variance is

$n$3

Finite-size scaling in experiment yielded $n$4, $n$5, $n$6, and $n$7, while MPS simulation gave $n$8, $n$9, $$|\psi\rangle \propto (I_n\otimes \langle s|)\,U_L\cdots U_2 U_1|0^m\rangle,$$0, and $$|\psi\rangle \propto (I_n\otimes \langle s|)\,U_L\cdots U_2 U_1|0^m\rangle,$$1. The first-moment statistical-mechanics model gave $$|\psi\rangle \propto (I_n\otimes \langle s|)\,U_L\cdots U_2 U_1|0^m\rangle,$$2, $$|\psi\rangle \propto (I_n\otimes \langle s|)\,U_L\cdots U_2 U_1|0^m\rangle,$$3, $$|\psi\rangle \propto (I_n\otimes \langle s|)\,U_L\cdots U_2 U_1|0^m\rangle,$$4, and $$|\psi\rangle \propto (I_n\otimes \langle s|)\,U_L\cdots U_2 U_1|0^m\rangle,$$5. All three approaches agreed on a classical random-walk universality class with diffusive dynamical scaling $$|\psi\rangle \propto (I_n\otimes \langle s|)\,U_L\cdots U_2 U_1|0^m\rangle,$$6 (Pokharel et al., 22 Sep 2025).

In the postselection-free $$|\psi\rangle \propto (I_n\otimes \langle s|)\,U_L\cdots U_2 U_1|0^m\rangle,$$7-symmetric setting, the entanglement transition is reconstructed from subsystem charge fluctuations rather than postselected trajectories. The protocol states that both the postselected entropy $$|\psi\rangle \propto (I_n\otimes \langle s|)\,U_L\cdots U_2 U_1|0^m\rangle,$$8 and the charge variance $$|\psi\rangle \propto (I_n\otimes \langle s|)\,U_L\cdots U_2 U_1|0^m\rangle,$$9 exhibit the same volume-law$A$00area-law transition, with critical measurement rate $A$01 and finite-size exponent $A$02. The reconstruction uses a universal mapping $A$03, described in the large-variance regime by a piecewise-linear form with $A$04 (Pöyhönen et al., 2024).

A different adaptive monitored transition is the absorbing-state transition in symmetric brickwork circuits. In the absence of symmetry-breaking noise, the long-time dynamics exhibits a transition at $A$05, accompanied by a jump in the intensive order parameter

$A$06

and a sharp peak in $A$07. With coherent symmetry breaking, the sharp phase boundary is rounded into a crossover band in the $A$08 plane (Ivaki et al., 2024).

4. Constant-depth state preparation and adaptive advantage

Adaptive monitored circuits have been used to prepare many-body states that are inaccessible to local unitary circuits of the same depth. A general constant-depth MPS construction proceeds by growing small MPS blocks in parallel, fusing them pairwise by Bell-type measurements, and then using classically conditioned local unitaries to push and cancel the random defects generated by the fusion outcomes. The relevant operator-pushing relation is

$A$09

so that virtual defects can be transferred to physical degrees of freedom and removed deterministically. The stated depth upper bound is 3 layers of entangling unitaries plus measurements plus 1 layer of feedforward unitaries, i.e. $A$10, independent of $A$11, correlation length $A$12, or fixed bond dimension $A$13 (Smith et al., 2024).

The set of preparable states described in that construction includes short- and long-ranged entangled MPS, symmetry-protected topological and symmetry-broken states, MPS with finite Abelian, non-Abelian, and continuous symmetries, resource states for MBQC, and families with tunable correlation length. Concrete examples include the AKLT state, GHZ states, Majumdar–Ghosh dimer states, and SPT families based on groups such as $A$14, $A$15, $A$16, and $A$17. The sufficient conditions emphasize injectivity after $A$18 blocking and the existence of appropriate on-site symmetries or finite-group defect bases (Smith et al., 2024).

The experimentally demonstrated depth advantage is clearest in the preparation of a toric-code ground state. A depth-4 adaptive circuit on a $A$19 rotated-surface-code strip uses ancilla-based stabilizer measurements, mid-circuit syndrome extraction, and classical feed-forward corrections to prepare the logical $A$20. The reported fidelity is at least $A$21, whereas any purely unitary depth-4 circuit on the same geometry and connectivity cannot prepare the target with fidelity greater than $A$22. The argument uses a locality-based bound derived from the non-overlap of past causal cones for distant logical operators (Foss-Feig et al., 2023).

This collection of results establishes a specific notion of adaptive advantage: measurements and classical feedforward alter the effective light-cone structure. In the MPS constructions, this appears as defect pushing and entanglement swapping; in the toric-code experiment, as constant-depth elimination of syndrome defects. The data support that adaptive monitored circuits can achieve tasks that constant-depth local unitary circuits cannot achieve under identical depth and connectivity constraints (Smith et al., 2024).

5. State complexity, ancilla trade-offs, and phase indicators

A recent complexity-theoretic analysis formalizes which states can be efficiently realized with limited ancilla and low depth in adaptive circuits. Two state properties are introduced: state weight $A$23 and anti-shallowness $A$24. The weight is defined from a lexicographically minimal maximal commuting generalized-stabilizer generating set and captures the maximal spatial range of correlation; the anti-shallowness is

$A$25

where $A$26 is the maximum overlap with any depth-$A$27 non-adaptive circuit. In the stated intuition, $A$28 correlation range, while $A$29 correlation strength (Liu et al., 21 Sep 2025).

The main trade-off theorems are explicit. Any depth-$A$30 adaptive circuit on $A$31 qubits that prepares $A$32 must satisfy

$A$33

and for any state $A$34 prepared by a depth-$A$35 adaptive circuit with $A$36 ancillas,

$A$37

The first bound yields a lower bound via state weight; the second yields a lower bound via anti-shallowness. The paper also states that good QLDPC code states can have $A$38 and be realized by shallow adaptivity with $A$39 ancillas by measuring $A$40 checks in parallel (Liu et al., 21 Sep 2025).

Illustrative examples show that $A$41 and $A$42 separate different resource regimes. For $A$43, one has $A$44 and $A$45; for $A$46, $A$47 and $A$48; for good QLDPC code states, $A$49 and $A$50. For the $A$51-qubit multi-controlled Toffoli, the associated magic state has $A$52, leading to the stated lower bound that any depth-$A$53 adaptive compilation requires either $A$54 or $A$55 (Liu et al., 21 Sep 2025).

The same quantities are proposed as phase indicators. If two states $A$56 and $A$57 are in the same phase in the sense of constant-depth non-adaptive connectivity, then both $A$58 and $A$59. In this sense, states whose $A$60 or $A$61 scale differently cannot lie in the same phase. The examples given distinguish product states from GHZ or toric-code states by $A$62, and distinguish GHZ from QLDPC code states by $A$63 (Liu et al., 21 Sep 2025).

6. Noise, hardware realization, simulation, and compilation

Adaptive monitored circuits have been implemented on superconducting hardware at scales well beyond early monitored-circuit experiments. The Bernoulli-map experiment used the “ibm_fez” Heron r2 superconducting processor with 156 fixed-frequency transmons and selected chains of up to $A$64 qubits to optimize $A$65, $A$66, and gate and readout error rates. The gate set comprised single-qubit $A$67 gates of $A$68 ns, two-qubit $A$69 gates of $A$70 ns, and mid-circuit measurement of $A$71 with conditional reset. Typical selected-qubit error rates were $A$72 for single-qubit gates, $A$73 for two-qubit gates, and $A$74 for measurement. Circuit sizes reached up to $A$75 $A$76 gates and $A$77 resets for $A$78, $A$79, with fifty random circuits and $A$80 shots per $A$81, and no post-selection or error mitigation (Pokharel et al., 22 Sep 2025).

Noise analysis in adaptive monitored circuits distinguishes coherent and incoherent symmetry breaking. In the symmetry-protected absorbing-state setting, single-qubit errors are modeled as random rotations

$A$82

inserted after two-qubit gates with probability $A$83, with $A$84 drawn uniformly from $A$85. The resulting error channel admits both Kraus and Lindblad descriptions and yields an average-gate-fidelity relation

$A$86

The reported conclusion is that coherent symmetry breaking rounds sharp transitions into crossovers, but adaptive feedback can still restore the ordered phase at sufficiently large measurement rate, with convergence time $A$87 persisting even for maximal noise $A$88 when $A$89 (Ivaki et al., 2024).

Classical analysis has developed along several complementary directions. For the Bernoulli-map transition, noiseless matrix-product-state simulation was performed with controlled truncation error $A$90 and bond dimension up to $A$91, together with noisy circuit simulation using single- and two-qubit depolarizing channels and a dephasing model, and replica statistical-mechanics mappings for first and second moments (Pokharel et al., 22 Sep 2025). For circuits with high rates of Pauli measurements and low non-Clifford content, a low-rank stabilizer-decomposition simulator represents the state as

$A$92

with systematic truncation by discarding coefficients $A$93. In the reported benchmarks, $A$94 sufficed to preserve observables to $A$95 accuracy, and in low-magic phases the method achieved polynomial memory and runtime growth where full state-vector simulation would require $A$96 resources (Aziz et al., 27 Aug 2025).

Compilation-oriented work has introduced a hypergraph representation for adaptive quantum circuits in which quantum lines and classical bits are vertices, while unconditional gates, measurements, conditional gates, and grouped logical modules become weighted hyperedges. In that representation, a measurement on qubit $A$97 introduces a measurement hyperedge $A$98, and a conditional gate depending on $A$99 becomes a hyperedge including both the acted-on qubits and the classical vertex. The partitioning problem is then posed as balanced hypergraph partitioning with an extended Fiduccia–Mattheyses heuristic that explicitly updates the cut costs of conditional and measurement hyperedges and can enforce super-hyperedge grouping for higher-level adaptive structures (Cambiucci et al., 12 Apr 2025).

7. Scope, limitations, and open directions

Across these developments, adaptive monitored quantum circuits occupy a boundary region between unitary circuit complexity, open-system dynamics, many-body entanglement, and hardware-level control. The current literature identifies several unresolved directions. For state-complexity theory, the bound $m=n+A$00 is stated as likely improvable to $m=n+A$01 in the generic unitary-plus-measure model, while approximate complexity in the intermediate-error regime remains open (Liu et al., 21 Sep 2025).

For tensor-network state preparation, the existing conditions are sufficient rather than necessary, and higher-dimensional PEPS present an obstruction because loops can trap defects; trees are straightforward, whereas MERA is identified as an open problem (Smith et al., 2024). For monitored entanglement experiments, the postselection-free $m=n+A$02 protocol still requires mid-circuit measurements, classical feedforward, and filtering to a charge sector, together with calibration of the parasitic volume-law coefficient $m=n+A$03 and the smoothing function $m=n+A$04 (Pöyhönen et al., 2024).

Noise robustness remains architecture-dependent. In the symmetry-protected absorbing-state setting, coherent noise eliminates sharp phase distinctions and replaces transitions by crossovers, even though states far from the original boundaries retain their essential character and adaptive feedback can suppress noise effects (Ivaki et al., 2024). In compilation and distributed execution, adaptive circuits require representations that track classical-control dependencies explicitly, and the adaptive hypergraph framework assumes a known probability distribution for measurement outcomes when weighting conditional edges (Cambiucci et al., 12 Apr 2025).

Taken together, the recent literature presents adaptive monitored quantum circuits not as a single protocol but as a class of measurement-feedback architectures. Their common feature is that mid-circuit observations are not passive diagnostics: they are integrated into the circuit law itself, altering reachable states, dynamical universality classes, complexity trade-offs, and experimental observables in ways that are unavailable to non-adaptive unitary evolution alone (Pokharel et al., 22 Sep 2025).