CQC

The Quantum Computing (QC) has been a topic of interest for quite a while now. However while the working prototypes exist, the general-purpose machines are decades away, even in the most optimistic scenario. In essence, QC is in the same state as the digital computing was back in 1950s.

At present, QC is in what is known as Noisy Intermediate-Scale Quantum (NISQ) era. The scaling in NISQ is limited to 1000 quantum gates, exceeding which the noise inherent in quantum circuits, accumulates to overwhelm the original signal rendering computation meaningless.

Our objective in forthcoming blogs, is to discuss the limitations in existing QC landscape, which must be overcome to take QC to the next level. The areas we will be interested in are:

1. Qubit architecture/Hardware: Moving beyond fragile qubits.
2. Algorithms: Designing for topological stability.
3. Software: Compiling for geometric structures.
4. Applications: Identifying where "shape-based" logic outperforms linear algebra.

We have already discussed QC at a very elementary level earlier. Almost a decade down the line, now our main focus however, is on what we think a logical computing really means in terms of manifolds. And to be very frank, we do not know what our findings will be in either of four categories mentioned above.

There are examples of highly sophisticated Topological Quantum Computers, which are the physical hardware built from topological phases. In this case the manifolds are braided together to complete a surface.

Cobordism allows us to mathematically approach this problem in a quite different manner. We want to understand how Cobordism relates two different boundary states and how it acts as the engine behind a logical computation. In a sense, we are more interested in the actual geometry behind manifolds and how it relates to the logical computation.

The fundamental equation behind Cobordic Quantum Computing (CQC) can be stated a follows:

"A quantum computation = A cobordism between Boundary Hilbert Spaces"

Boundary Hilbert Spaces are the vector spaces of quantum states that live on the edge (boundary) of a physical region. For example, gravity in a volume (the bulk) is mathematical dual to a quantum field theory living on its boundary.

However, the important point to note is that the boundary itself is not a conventional 1-dimensional boundary. The boundary can be either 3D or n-dimensional whereas the bulk is then 4D or n+1-dimensional respectively. In j-space, the "bulk" is the logic while the "boundary" is the data or observable(s), as measured by a finite capability observer (

α

< 1).¹

The simplest example to visualize CQC mechanism, is the event-horizon of a black-hole. In Beckenstein-Hawkings view, all the information contained within a black-hole (3D bulk) is encoded on its 2D surface i.e. event-horizon. The event-horizon is the null surface defined by the light-like (null) geodesics.

As another example, we can think of a soap bubble, where the surface is the thin-film and bulk is represented by volume of the air trapped inside. In conventional QC, the thin film is manipulated bit-by-bit using time-evolution. However in CQC, the bulk is mathematically constrained to change, based on the information placed on the surface.

Replace this picture of bulk with space-time. When the information on the surface is changed, the space-time is mathematically constrained (cobordism) to move to next state as the information on the surface and the bulk must be aligned to satisfy GR.² And this is the mechanism behind the logical computation in CQC. We note that in essence we are constraining the manifold to permissible solutions in space-time.

A summary of the comparison between standard Quantum Computing and CQC is provided below:

The fundamental idea behind the algebraic geometry is that, "All the geometric information of a space can be encoded in the algebraic structures." The simplest example is that a circle (geometry) can be written as

{x}^{2}+{y}^{2}=1

(algebra).

In topology, we enhance this idea to include the "connectivity" of the "spaces". Before we write algebra, we must decide if the shapes are topologically equivalent. Often quoted example, is that a doughnut and a coffee mug are topologically equivalent to each other as their Euler characteristics are the same. This enhancement is extremely important as we study the underlying structure (bulk) before we launch into writing complicated equations (boundaries). (Arguably, the algebraic equation of a torus is much simpler than that for a coffee mug.)

We can think of CQC as the following categories:

1. Physical hardware built from topological phases
2. Logical computation defined as cobordism between boundary states

The category-1 refers to, refers to quantum computing platforms where information is stored and manipulated using the global topological properties of a material, rather than fragile local states like individual electron spins or photon polarizations.

In certain exotic materials—such as topological superconductors or fractional quantum Hall systems—quasiparticles called anyons can appear. When these anyons are moved (or braided) around each other, the system’s quantum state changes in a way that depends only on the topological pattern of the braid, not on the exact path taken or small environmental disturbances. Because topology is insensitive to small errors, this approach naturally protects quantum information from noise and decoherence.

In practice, the “hardware” consists of engineered materials and nanostructures where these topological excitations can be created, moved, and measured, turning braiding patterns into quantum logic operations. This idea underlies topological quantum computing, pursued experimentally in systems like fractional quantum Hall devices, Majorana zero modes in superconducting nanowires, and other strongly correlated quantum materials.

However, our focus is on CQC category-2. The logical computation defined as cobordism between boundary states, is a viewpoint where a computation is not described as a sequence of time-ordered gates acting on qubits, but as a single geometric object connecting an initial state and a final state.

In topology, a cobordism is a higher-dimensional space whose boundary consists of two lower-dimensional shapes. Translating this idea to computation, the input state and output state are treated as boundaries, and the entire computation is the bulk structure that connects them. The logical transformation is therefore determined by the global properties of this connecting space rather than by step-by-step operations. A simple example is a cylinder whose two ends are connected through a higeher dimensional bulk.

The simplest Cobordic Quantum Computer

Different possible “fillings” or geometries between the same boundaries can contribute phases or transformations, much like path integrals in quantum physics. This perspective shifts the focus from local gate operations to global topology, allowing computation to be understood as a relationship between boundary states mediated by an allowed geometric bridge.

We will be discussing next, some basic concepts related to the Cobordic Quantum Computing in the discrete measurement space or j-space. It will be followed by Michael Atiyah axioms for a Topological Quantum Field Theory (TQFT), which formalize how cobordisms act as “processes” connecting spaces. The key idea is that "a TQFT assigns algebraic objects to spaces and linear maps to cobordisms and therefore geometry $→$ algebra."

....to be continued

______________

1. The examples of elements of boundaries are DNA and Knot polynomials. We can also say that the boundaries are where the stable structures (observables) live.

2. The geometric quantities on the LHS of Einstein's equations (the Einstein tensor

{G}_{μν}

) are effectively encodings of the information density stored in the Boundary Hilbert Space (

{S}_{0}


	Cobordic Quantum Computing: Defining Logical Computation through Manifold Geometry Beyond NISQ 29^th March 2026	HOME
QUOTES “Today the physicist sacrifices structural stability for computability. I hope that he will not have cause to regret this choice." — René Thom (The Founder of Cobordism) "We can view any cobordism M between S₀and S₁ as inducing a linear transformation between their respective Hilbert spaces." - Michael Atiyah (Cobordism in Quantum Field Theory) “It is very possible that a proper understanding of string theory will make the space-time continuum melt away.” — Edward Witten (Cobordism and the Nature of Reality) “It is most natural to suppose that the entropy of a black hole is proportional to the area of its horizon.” “The entropy of a black hole measures the amount of information about the physical state of matter which is hidden from an outside observer.” — Jacob Bekenstein	The Quantum Computing (QC) has been a topic of interest for quite a while now. However while the working prototypes exist, the general-purpose machines are decades away, even in the most optimistic scenario. In essence, QC is in the same state as the digital computing was back in 1950s. At present, QC is in what is known as Noisy Intermediate-Scale Quantum (NISQ) era. The scaling in NISQ is limited to 1000 quantum gates, exceeding which the noise inherent in quantum circuits, accumulates to overwhelm the original signal rendering computation meaningless. Our objective in forthcoming blogs, is to discuss the limitations in existing QC landscape, which must be overcome to take QC to the next level. The areas we will be interested in are: 1. Qubit architecture/Hardware: Moving beyond fragile qubits. 2. Algorithms: Designing for topological stability. 3. Software: Compiling for geometric structures. 4. Applications: Identifying where "shape-based" logic outperforms linear algebra. We have already discussed QC at a very elementary level earlier. Almost a decade down the line, now our main focus however, is on what we think a logical computing really means in terms of manifolds. And to be very frank, we do not know what our findings will be in either of four categories mentioned above. There are examples of highly sophisticated Topological Quantum Computers, which are the physical hardware built from topological phases. In this case the manifolds are braided together to complete a surface. Cobordism allows us to mathematically approach this problem in a quite different manner. We want to understand how Cobordism relates two different boundary states and how it acts as the engine behind a logical computation. In a sense, we are more interested in the actual geometry behind manifolds and how it relates to the logical computation. The fundamental equation behind Cobordic Quantum Computing (CQC) can be stated a follows: "A quantum computation = A cobordism between Boundary Hilbert Spaces" Boundary Hilbert Spaces are the vector spaces of quantum states that live on the edge (boundary) of a physical region. For example, gravity in a volume (the bulk) is mathematical dual to a quantum field theory living on its boundary. However, the important point to note is that the boundary itself is not a conventional 1-dimensional boundary. The boundary can be either 3D or n-dimensional whereas the bulk is then 4D or n+1-dimensional respectively. In j-space, the "bulk" is the logic while the "boundary" is the data or observable(s), as measured by a finite capability observer ( $α$ < 1).¹ The simplest example to visualize CQC mechanism, is the event-horizon of a black-hole. In Beckenstein-Hawkings view, all the information contained within a black-hole (3D bulk) is encoded on its 2D surface i.e. event-horizon. The event-horizon is the null surface defined by the light-like (null) geodesics. As another example, we can think of a soap bubble, where the surface is the thin-film and bulk is represented by volume of the air trapped inside. In conventional QC, the thin film is manipulated bit-by-bit using time-evolution. However in CQC, the bulk is mathematically constrained to change, based on the information placed on the surface. Replace this picture of bulk with space-time. When the information on the surface is changed, the space-time is mathematically constrained (cobordism) to move to next state as the information on the surface and the bulk must be aligned to satisfy GR.² And this is the mechanism behind the logical computation in CQC. We note that in essence we are constraining the manifold to permissible solutions in space-time. A summary of the comparison between standard Quantum Computing and CQC is provided below: The fundamental idea behind the algebraic geometry is that, "All the geometric information of a space can be encoded in the algebraic structures." The simplest example is that a circle (geometry) can be written as ${x}^{2}+{y}^{2}=1$ (algebra). In topology, we enhance this idea to include the "connectivity" of the "spaces". Before we write algebra, we must decide if the shapes are topologically equivalent. Often quoted example, is that a doughnut and a coffee mug are topologically equivalent to each other as their Euler characteristics are the same. This enhancement is extremely important as we study the underlying structure (bulk) before we launch into writing complicated equations (boundaries). (Arguably, the algebraic equation of a torus is much simpler than that for a coffee mug.) We can think of CQC as the following categories: 1. Physical hardware built from topological phases 2. Logical computation defined as cobordism between boundary states The category-1 refers to, refers to quantum computing platforms where information is stored and manipulated using the global topological properties of a material, rather than fragile local states like individual electron spins or photon polarizations. In certain exotic materials—such as topological superconductors or fractional quantum Hall systems—quasiparticles called anyons can appear. When these anyons are moved (or braided) around each other, the system’s quantum state changes in a way that depends only on the topological pattern of the braid, not on the exact path taken or small environmental disturbances. Because topology is insensitive to small errors, this approach naturally protects quantum information from noise and decoherence. In practice, the “hardware” consists of engineered materials and nanostructures where these topological excitations can be created, moved, and measured, turning braiding patterns into quantum logic operations. This idea underlies topological quantum computing, pursued experimentally in systems like fractional quantum Hall devices, Majorana zero modes in superconducting nanowires, and other strongly correlated quantum materials. However, our focus is on CQC category-2. The logical computation defined as cobordism between boundary states, is a viewpoint where a computation is not described as a sequence of time-ordered gates acting on qubits, but as a single geometric object connecting an initial state and a final state. In topology, a cobordism is a higher-dimensional space whose boundary consists of two lower-dimensional shapes. Translating this idea to computation, the input state and output state are treated as boundaries, and the entire computation is the bulk structure that connects them. The logical transformation is therefore determined by the global properties of this connecting space rather than by step-by-step operations. A simple example is a cylinder whose two ends are connected through a higeher dimensional bulk. The simplest Cobordic Quantum Computer Different possible “fillings” or geometries between the same boundaries can contribute phases or transformations, much like path integrals in quantum physics. This perspective shifts the focus from local gate operations to global topology, allowing computation to be understood as a relationship between boundary states mediated by an allowed geometric bridge. We will be discussing next, some basic concepts related to the Cobordic Quantum Computing in the discrete measurement space or j-space. It will be followed by Michael Atiyah axioms for a Topological Quantum Field Theory (TQFT), which formalize how cobordisms act as “processes” connecting spaces. The key idea is that "a TQFT assigns algebraic objects to spaces and linear maps to cobordisms and therefore geometry $→$ algebra." ....to be continued ______________ 1. The examples of elements of boundaries are DNA and Knot polynomials. We can also say that the boundaries are where the stable structures (observables) live. 2. The geometric quantities on the LHS of Einstein's equations (the Einstein tensor ${G}_{μν}$ ) are effectively encodings of the information density stored in the Boundary Hilbert Space ( ${S}_{0}$ ).	Previous Blogs: TM-Kaluza's Theory Part-2 TM-Kaluza's Theory Part-1 TM-Physics in ACTION TM-Fine Structure Constant TM-Frames of References TM-State of Measurement The Theory of Measurements - Intro An Ecosystem of δ-Potentials - IVB An Ecosystem of δ-Potentials - IVA An Ecosystem of δ-Potentials - III An Ecosystem of δ-Potentials - II An Ecosystem of δ-Potentials - I Nutshell-2019 Stitching Measurement Space - III Stitching Measurement Space - II Stitching Measurement Space - I Mass Length & Topology A Timeless Constant Space Time and Entropy Nutshell-2018 Curve of Least Disorder Möbius & Lorentz Transformation - II Möbius & Lorentz Transformation - I Knots, DNA & Enzymes Quantum Comp - III Nutshell-2017 Quantum Comp - II Quantum Comp - I Insincere Symmetry - II Insincere Symmetry - I Existence in 3-D Infinite Source Nutshell-2016 Quanta-II Quanta-I EPR Paradox-II EPR Paradox-I De Broglie Equation Duality in j-space A Paradox The Observers Nutshell-2015 Chiral Symmetry Sigma-z and I Spin Matrices Rationale behind Irrational Numbers The Ubiquitous z-Axis
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Cobordic Quantum Computing: Defining Logical Computation through Manifold Geometry

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