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Figure 5: Comparison with existing metrics on Motion-X. Existing evaluation metrics (Guo et al., 2022) are illustrated in red, and ours are in green. The B = 32 and B = 256 settings for retrieval are denoted as “ •−−” and “ ▲−−” respectively.

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Figure 6: Sentences similarity comparison between the sBERT and MPNet .

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Figure 7: Visualization of MotionCLIP generated results. The first frame and the final frame of motions are shown in the figure.

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Figure 8: Qualitative comparison with T2M-GPT.

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Figure 9: Visualization of the whole-body motions generated by HumanTOMATO (1).

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Table 1: Main results of motion generation on Motion-X dataset.

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Table 10: Different vector quantization methods on Motion-X.

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Table 11: Different vector quantization methods on GRAB.

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Table 12: Different vector quantization methods on HumanML3D.

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Table 13: The ablation on how can H2VQ help the whole-body motion generation on T2M-GPT.

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Table 14: Abaltion on how pre-trained text-motion aligned model helps to generate the text-aligned body-only motion (on HumanML3D).

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Table 15: R-Precision of GT motions and texts on the Motion-X dataset.

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Table 16: R-Precision of GT motions and texts on the HumanML3D dataset.

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Table 2: Comparison of the motion reconstruction errors (MPJPE in mm) of different quantization methods on Motion-X, GRAB, and HumanML3D. Our H2VQ shows significant improvements.

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Table 3: Abaltion on a pre-trained text-motion-aligned model for motion generation on Motion-X. Both TMR embedding and text-motion alignment supervision help generate text-aligned motions.

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Table 4: Ablation study of different motion representations on the Humanml3D dataset.

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Table 5: Recall@K (T2M and M2T) of GT motions and texts on the Motion-X dataset.

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Table 6: Recall@K (T2M and M2T) of GT motions and texts on the HumanML3D dataset.

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Table 7: Quantitative Comparison on the Motion-X dataset (FID, TMR-R-Precision(256), and RPrecision(32) metrics).

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Table 8: Quantitative Comparison on the Motion-X dataset (TMR-Matching Score, Matching Score, and MModality metrics).

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Table 9: Facial motion generation results on Motion-X dataset.

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Figure 1: Standard SymTFT setup. Topological symmetry operators (green, U) link heavy defect operators (grey, D) in the (D + 1)-dimensional slab. The defects stretch from the topological boundary (blue, Btop) to the physical boundary (red, Bphys).

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Figure 10: Sketch of double throat internal geometry with two sets of localized degrees of freedom (red). Horizontal slices of constant radius are initially disjoint and then combine resulting in a connected asymptotic boundary.

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Figure 12: (i): String / brane running between two local models. Such an object admits a partition into three pieces, two of which are contained in a local model, and one which connect the two via the bulk of X. In the QFT spacetime the result is a pair of defects E(12). (ii): Deforming the configuration into a spacetime direction x⊥ the bulk part of the string / brane gives a topological operator bounded by the defects. Individually the defects E(i) are non-genuine.

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Figure 15: Pair of pants describing the uplift of a SymTree junction.

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Figure 17: We sketch the partially smoothed geometry X ′ as a fibration over a Y-shaped base. This picture presents a horizontal slice of figures similar to figure 2.

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Figure 18: Depiction of line operators after adjoint Higgsing of 4D N = 4 SYM with gauge group SU(N) = SU(N1 +N2) to S(U(N1)× U(N2)). A non-genuine ’t Hooft line (0, 1) and dyonic line (1,−1) of two sectors of the multi-sector theory can fuse at the junction with an electric Wilson line (1, 0). These are realized by (p, q)-strings, i.e., bound states of p F1-strings and q D1-strings. This implements an explicit example of the general phenomena anticipated in figure 11.

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Figure 2: We depict a trivalent junction J of symmetry TFTs. The junctions supports the D-dimensional theory GJ ⊗ TFTJ . Color conventions: Junctions are purple.

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Figure 20: SymTree with topological boundary condition for SU(N1 + N2 + · · · + NK) global form, and junction and tree Υ describing the Higgsing SU(N1 + N2 + · · · + NK) ⊃ S(U(N1)×U(N2)×· · ·×U(NK)). Node theories are 4D SYM with indicated gauge algebra.

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Figure 22: SymTrees for brane probes of singularities. (i): stack of K D3-branes probing an C2/ZN singularity. In the extra dimensional geometry the D3-branes and the singularity are separated. (ii): cross-section of this configuration. (iii): we contract a branch / center the filtration on the D3-brane stack. Moving radially outwards from the D3-brane stack the linking S5 K with K units of D3 flux sweeps over the singularity and is folded to an (S5/ZN)K . (iv): we contract another branch pushing the brane stack into the singularity. This alters the physical boundary condition and the massless edge modes are now organized into a quiver gauge theory. The shaded slabs / red cross-section signify, from the perspective of 4D edge modes / 5D slabs, that the ADE locus is non-compact. It stretches to infinity of the respective filtration and alters the boundary conditions. (v): the SymTree of two stacks of D3-branes probing a partial resolution of a C2/ZN singularity. The internal dimensions contain a pair C2/ZNi singularities and Ki D3-brane, where i = 1, 2, and which are all separated. In (vi) we show the cross-section and label branches by the geometry of the corresponding radial shells and their D3-brane flux. Note, there is a purely geometric junction J (Sing) and junction purely characterized by adjoint Higgsing of a brane stack J (D3).

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Figure 23: We depict the base of the local K3 π : X → B = C. The preimage under π of a base filtration FB gives a filtration FX . The critical slice projects to a figure eight. The filtration for the base lifts to the full geometry, as determined by the SL(2,Z) monodromy matrices M1,M2.

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Figure 24: SymTree derived from the filtration FX of the orbifold X = (T 2 × C2)/Z3. We give the topological models for the radial shells at the legs and junctions. The junction valency is 4.

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Figure 25: On the left, we depict the collection of separated 4D N = 4 suNi SYM theories that we label by Ti where Ni = M + ϵi and a sum of local operators in each of these sectors. The wedges represent the AdS5 × S5 dual spacetimes for each sector. On the right, we illustrate the averaged operator O in the ensemble averaged theory T .

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Figure 26: In the SymTFT frame work changing from SU(2) to SU(2)/Z2 = SO(3)+ gauge theory can be formulated as a change in topological boundary condition. Equivalently, we can realize this as an insertion of the Fourier operator P . This operator can the be collided with the physical boundary condition giving a notion of coupling the relative physical boundary to a TFT.

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Figure 27: SymTree with two topological boundary conditions realizing the polarization change SU(2) to SO(3)+.

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Figure 28: Sketch of the Hopf-fibration S3/ZNi → S2 and the bounding chain Σi within it. The Euler class NivolS2 of this circle fibration characterizes the obstruction to the existence of a section. Consider attempting to construct such a section, as depicted, by starting at the south pole of S2 and growing a disk inside of S3/ZNi , projecting to S2 as shown. Upon reaching the North pole the boundary ∂Σi does not close, rather it winds Ni times around the Hopf fiber S1 H . With this Σi is a chain bounding Ni copies of S 1 H .

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Figure 29: Depict the covering of the large radius Mayer-Vietoris sequence with respect to the M-theory circle fibration. In R3, the IIA dual to X ′, we have two spheres S2 i touching along a two-disk D2 marked blue. The preimage of this disk and its complement are the large radius covering. The boundary of the preimage of the disk, which is the intersection of the two covering sets, is the circle S1 H fibered over the boundary of the disk.

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Figure 3: Junctions can be assembled into trees (i). The tree Υ can be visualized as a horizontal cross-section. Junctions can have arbitrary valency (ii).

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Figure 30: Sketch of the geometry X◦.

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Figure 4: Depiction of retracting a SymTree to produce the corresponding SymTFT Sfull for the multi-sector QFT with topological couplings between the different sectors. In terms of the SymTree, this amounts to pulling in the different branches into the physical boundaries.

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Figure 6: We depict two SymTrees related by an associator move Υ ↔ Υ′. There is an anomaly whenever fusion of trivalent junctions produces distinct tetravalent junctions. Generalizations are immediate. This can be accompanied by a non-zero obstruction class / anomaly αΥ,Υ′ , although the examples considered in this paper have no such obstruction.

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Figure 7: Sketch of possible defect configurations for a trivalent junction of symmetry TFTs. The red and purple dots denote the spacetime defects, the brown line marks the defect within the SymTree. The purple junction defect is said to dress the red boundary defects.

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Figure 8: We sketch a deformation of the defect configuration depicted in (i). In (i) the purple and red defect are coincident in spacetime, as shown on the lefthand side in (iii). In (ii) we displace these along a spacetime direction x⊥ and deforming the resulting defect configuration into a horizontal and vertical piece we find a topological operator bounded by the initial pair of defects, as depicted on the righthand side in (iii).

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Figure 9: Sketch of different topological symmetry operator configurations for a trivalent junction of symmetry TFTs. We consider the initial configuration depicted in (i). Deforming it (partially) across/into the junction gives various equivalent configurations of topological operators presented in subfigures (ii) - (vi). The dashed lines indicate topological operators in one higher dimension. We denote non-genuine operators at their boundaries as V ,V(1),V(2), also represented by green dots. The purple dots again depict dressings. Both the dressings and the higher-dimensional operators can be trivial.

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FIG. 1. A rectangular lattice of qubits used for a pairwise measurement-based realization of the surface code. Data qubits are shown as open dots and auxiliary qubits are shown as solid dots. The blue and red squares will correspond to Z-type and X-type plaquettes (4-gons), respectively. Each plaquette exclusively utilizes three auxiliary qubits (labeled A-C) to execute its stabilizer measurement circuit.

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FIG. 10. The possible reductions of Z-type n-gons for step 1, where dead data qubits (denoted by starbursts) are removed from the code operation. (Reductions related to these by rotations and reflections are not shown separately.) The reductions for X-type n-gons may be obtained from these by 90 degree rotations. Even though this step only considers dead data qubits, the auxiliary qubits are displayed to show the collateral removal of live auxiliary qubits that may occur when removing dead data qubits. Ignoring the auxiliary qubits, the same reduction of n-gons can be used for any realization of the surface code. The full reduction of each n-gon in this step can be determined through an iterative process where dead data qubits are removed one at a time until no dead ones remain, i.e working down through this figure.

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FIG. 11. The possible splittings of Z-type n-gons for step 2, where dead auxiliary qubits (denoted by starbursts) are removed from the code operation. (Splittings related to these by rotations and reflections are not shown separately.) The splittings for X-type n-gons may be obtained from these by 90 degree rotations. In some cases, live auxiliary qubits will be collaterally removed as a result of removing dead auxiliary qubits. The full splitting of each n-gon in this step can be determined through an iterative process where dead auxiliary qubits are removed one at a time (together with any collateral loss), until no dead ones remain, i.e working down through this figure.

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FIG. 12. The possible splittings of Z-type n-gons for step 3, where dead connections (denoted by starbursts) are removed from the code operation. (Splittings related to these by rotations and reflections are not shown separately.) The splittings for X-type n-gons may be obtained from these by 90 degree rotations. In some cases, live auxiliary qubits will be collaterally removed as a result of removing dead connections. The full splitting of each n-gon in this step can be determined through an iterative process where dead connections are removed one at a time (together with any collateral loss), until no dead ones remain, i.e working down through this figure.

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FIG. 13. Configuration for one dead data qubit.

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FIG. 18. Measurement loops in Majorana hardware corresponding to the measurements needed for our code. MZMs (red circles) are located at the end points of topological wires (gray lines). Two topological wires connected by a trivial superconducting spine (dark gray) form a tetron. Tetrons can be connected to each other or coherent links through semiconductor segments (tan).

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FIG. 19. Interference loops of measurements for the implementation of our measurement-based surface code realization in Majorana hardware with double-rail semiconductor layouts. This layout avoids loop conflicts, allowing for the (minimal) period 4 operation schedule.

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FIG. 2. The MZZZZ circuit for measuring ZZZZ on four data qubits using three auxiliary qubits. The data qubits are labeled 1-4 in the order that they are addressed in this circuit. The auxiliary qubits are labeled A-C. The measurement schedule shown here is useful for repeatedly applying the measurement circuit with a four step period, which can be done by repeating steps 1-4.

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FIG. 20. The implementation of our measurement-based surface code realization in Majorana hardware with single-rail semiconductor layouts requires resolution of loop conflicts in two steps. A resolution with period 5 operation schedule is shown here.

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FIG. 21. (left) Performance results for our pairwise measurement-based surface code realization described in Sec. II on a rotated surface code patch using boundary conditions that make the hook errors benign (see Sec. IV). (right) Performance results for the 4.8.8 Floquet code on a planar patch with rectangular boundary conditions (as in Ref. 5). For implementation in Majorana hardware, these require double-rail semicondutor layouts. Fault-tolerance thresholds are found to be 0.66% for our code and 1.3% for the 4.8.8 Floquet code.

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FIG. 22. Resource requirements to reach target logical error rates of ptargetlogical = 10−8, 10−12, and 10−15, comparing our realization of the surface code and the 4.8.8 planar Floquet code. The respective target logical error rates correspond to columns from left to right, and the resource quantities being considered correspond to rows, from top to bottom: qubit count, circuit depth, and spacetime footprint (i.e., qubit count times circuit depth).

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FIG. 23. (left) Performance results for the single-rail variant of our pairwise measurement-based surface code realization described in Sec. VI on a rotated surface code patch using boundary conditions that make the hook errors benign. (right) Performance results for the single-rail variant of 4.8.8 Floquet code on a planar patch with rectangular boundary conditions (see description in text). These variants can be implemented in Majorana hardware with single-rail semicondutor layouts. Fault-tolerance thresholds for these single-rail variants are found to be 0.51% for our code and 0.52% for the 4.8.8 Floquet code.

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FIG. 24. Resource requirements to reach target logical error rates of ptargetlogical = 10−8, 10−12, and 10−15, comparing single-rail variants of our realization of the surface code and the 4.8.8 planar Floquet code.

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FIG. 25. (left) Performance results for our pairwise measurement-based surface code realization on a rotated surface code patch using boundary conditions that make the hook errors malignant (see Sec. IV). We note that the d = 3 curve is not expected to intersect with the other curves at (or near) threshold, as the code is not error-correcting at d = 3, because the the fault distance is df = 2. (right) Performance results for our code on a torus. Fault-tolerance thresholds are found to be 0.65% for the planar patch and 0.70% for the torus.

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FIG. 26. Performance results for the hook-preventing variant of our pairwise measurement-based surface code realization described in Sec. III on a torus for noise model without (left) and with (right) idle noise. The full code distance is recovered and performance is improved with respect to the code using the original circuits when hook errors are malignant (see Fig. 25). Fault-tolerance thresholds are found to be 0.61% where there are no idle errors and 0.43% when idle errors are included.

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FIG. 27. The stabilizer measurement circuits can be interleaved and run on a synchronous schedule, i.e. (0Z, 0X), (1Z, 1X), . . ., by addressing the data qubits in a different order than the pipelined measurement schedule presented in Sec. II. This interleaved measurement schedule has significant disadvantages compared to the pipelined schedule.

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FIG. 28. The MZZZZ circuit from Ref. 9 for the pentagonal tiling realization of the surface code.

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FIG. 29. A modification of the MZZZZ circuit from Ref. 9 that prevents the problematic hook errors associated with readout error at the MXAXB measurement. Incorporating this and a similar modification of the MXXXX circuit reduces the problem of bidirectional hook errors to unidirectional hook errors in the pentagonal tiling realization of the surface code.

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FIG. 30. The possible splittings of Z-type n-gons for step 2, where dead auxiliary qubits are removed from the code operation for the measurement-based pentagonal tiling realization of the surface code. (Splittings related to these by rotations and reflections are not shown separately.) The splittings for X-type n-gons may be obtained from these by 90 degree rotations.

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FIG. 31. The possible splittings of Z-type n-gons for step 3, where dead connections are removed from the code operation for the measurement-based pentagonal tiling realization of the surface code. (Splittings related to these by rotations and reflections are not shown separately.) The splittings for X-type n-gons may be obtained from these by 90 degree rotations.

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FIG. 32. The splittings of Z-type or X-type plaquettes for steps 2 and 3, where dead auxiliary qubits and connections are removed from the code operation for the CNOT gate-based realization of the surface code. All possible n-gons splittings are not shown because they all follow the same pattern: for step 2, a dead auxiliary qubit splits the n-gon into n 1-gons; for step 3, a dead connection splits the n-gon into a (n−1)-gon and a 1-gon, according to which connection is dead.

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FIG. 4. The MZZZZ and MXXXX measurement circuit steps for Z-type and X-type plaquettes. Labels indicate the operators measured on the respective qubits in the given step, with connecting lines indicating pairwise measurement. These circuits can be applied in parallel with a relative shift in their operation schedules. A 4 step period for repeated application of these surface code stabilizer measurement circuits is obtained using the pipelining: · · · , (1Z, 3X), (2Z, 4X), (3Z, 1X), (4Z, 2X), · · · . Steps 1-4 of a given circuit are applied repeatedly, whereas steps 0 and 5 are only used in the ramp up and down of the repetition of circuits, as indicated in Eqs. (1)-(3).

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FIG. 5. An example of a hook error in the MZZZZ measurement circuit is given by a Z error on auxiliary qubit B that occurs between steps 2 and 3, as shown in the circuit on the left. This error is equivalent to a ZZ error on data qubits 1 and 3, as shown in the circuit on the right. (Errors are marked in yellow.)

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FIG. 6. The modified circuit for MZZZZ in which there are no hook errors. Steps 3, 4, and 7 shown here are the additional steps that have been inserted into the original circuit from Fig. 2. (The MZB auxiliary qubit measurement only needs to occur twice per cycle, but we have shown it occurring three times.) Here, we only show the steps for one round of repeated application of the measurement circuit. In order to ramp up the circuit, one needs to begin with a step 0 that applies a MXA measurement before step 1; this could be achieved (with additional redundant measurements) by applying step 7.

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FIG. 7. The hook-preventing MZZZZ and MXXXX measurement circuit steps for Z-type and X-type plaquettes. These can be pipelined as in Eq. (4) to achieve a 7 step period.

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FIG. 8. Z-type (left) and X-type (right) measurement circuits for 1-gons, 2-gons, and 3-gons.

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FIG. 9. Different boundary conditions for a square patch of surface code, all shown for fault distance df = 3. (a) The rotated surface code with a good choice of boundary conditions aligns the logical strings perpendicular to the corresponding hook errors of the same type, so df = d. The relation between number of physical qubits and fault distance in this case is N = 4d2f − 4df + 1. (b) The rotated surface code with a bad choice of boundary conditions aligns the logical strings parallel to the corresponding hook errors of the same type, which halves the fault distance, so df = ⌈d/2⌉. The resulting relation between number of physical qubits and the fault distance in this case is N = 16d2f + 4df − 5. (c) The original (unrotated) surface code has the logical strings aligned diagonal to the direction of the hook errors, so df = d. The relation between number of physical qubits and fault distance in this case is N = 8d2f − 8df + 1.

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TABLE I. Comparison between key properties of pairwise measurement-based codes, including our realization of the surface code (denoted “3aux”), the double ancilla and windmill realizations of Ref. 8, the pentagonal tiling realization of Ref. 9, and the Floquet code on the 4.8.8 and honeycomb lattices [5, 10–12]. The term “single-rail” indicates variants of the corresponding codes that are needed to make them compatible with Majorana hardware using single-rail semiconductor layouts. Total qubit count for a logical patch is given to leading order as function of fault distance df . Circuit depth is given per round of syndrome extraction. Fault tolerance thresholds are computed with respect to the noise model of Ref. 8, except for that of the pentagonal tiling realization from Ref. 9 (indicated by ∗), which uses a slightly different noise model. We indicate whether the code can be implemented in Majorana hardware using simple layouts and measurements, or if it requires complicated layouts and measurements that are likely prohibitive.

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TABLE II. Fault distance df required to reach a target logical error rate ptargetlogical of 10 −8, 10−12 and 10−15, respectively, comparing our realization of the surface code and the 4.8.8 planar Floquet code. The determination of the required df is the same as in Fig. 22.

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TABLE III. Fault distance df required to reach a target logical error rate ptargetlogical of 10 −8, 10−12 and 10−15, respectively, comparing single-rail variants of our realization of the surface code and the 4.8.8 planar Floquet code.

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Figure 2. Overview of Cutie. We store pixel memory F and object memory S representations from past segmented (memory) frames. Pixel memory is retrieved for the query frame as pixel readout R0, which bidirectionally interacts with object queries X and object memory S in the object transformer. The L object transformer blocks enrich the pixel feature with object-level semantics and produce the final RL object readout for decoding into the output mask. Standard residual connections, layer normalization, and skip-connections from the query encoder to the decoder are omitted for readability.

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Figure 3. Visualization of cross-attention weights (rows of AL) in the object transformer. The middle cat is the target object. Top: without foreground-background masking – some queries mix semantics from foreground and background (framed in red). Bottom: with foreground-background masking. The leftmost three are foreground queries, and the remaining are background queries. Semantics is thus cleanly separated. The f.g./b.g. queries can communicate in the subsequent self-attention layer. Note the queries attend to different foreground regions, distractors, and background regions.

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Table 1. Quantitative comparison on video object segmentation benchmarks. All algorithms with available code are re-run on our hardware for a fair comparison. We could not obtain the code for [67, 68, 75] at the time of writing, and thus they cannot be reproduced on datasets that they do not report results on. For a fair comparison, all methods in this table use ImageNet [89] pre-training only or are trained from scratch. We compare methods with external training (e.g., MAE [69] pre-training) in the supplement. ∗estimated FPS.

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Table 2. Comparisons of performance on long videos on the BURST dataset [3]. Mem.: maximum GPU memory usage. FIFO: first-in-first-out memory bank; INF: unbounded memory; LT: longterm memory [9]. DeAOT [66] is not compatible with long-term memory. All methods are trained with the MOSE [12] dataset.

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Table 5. Ablations on positional embeddings in the object transformer.

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