Representation Theory And Its Surprising Applications In Complexity Theory

Understanding Representation Theory

Representation theory is the study of abstract algebraic structures by representing their elements as linear transformations of vector spaces. Some key concepts in representation theory include:

• Group representations – representing elements of a group as matrices so that the group operation corresponds to matrix multiplication.
• Character theory – studying traces of group representations to understand their structure.
• Modular representations – representations over finite fields which have applications in complexity theory.
• Rigidity – a property of some representations that makes them useful for proving circuit lower bounds.

Despite its abstract nature, representation theory has found surprising connections and applications in complexity theory. The algebraic tools of representation theory provide new techniques for tackling the fundamental problems of proving circuit lower bounds and derandomization.

Applications of Representation Theory in Complexity Theory

Proving Circuit Lower Bounds

Proving superpolynomial lower bounds on circuit size for explicit functions is a notoriously difficult open problem in complexity theory. Representation theory has offered glimmers of progress in this area through two main approaches:

• The polynomial method utilizes the properties of specific group representations over finite fields to prove circuit lower bounds. The best known bound from this technique is on lower bounds for arithmetic circuits due to Gupta et al. utilizing representations of symmetric groups.
• Rigid matrix method relies on the rigidity property of certain matrix-valued functions consisting of irreducible representations over fields. Razborov first discovered a connection between rigid matrices and circuit complexity. Smolensky further extended this method to prove lower bounds for bounded depth arithmetic circuits. However, major barriers remain in extending these bounds to general circuits.

While significant progress remains to be made, representation theory offers a toolkit of algebraic techniques that shed new light on the venerable challenge of proving circuit lower bounds.

Derandomization

Representation theory has also yielded tools for derandomizing algorithms, which aim to minimize or eliminate their use of randomness. A major breakthrough was made by Impagliazzo and Wigderson, who established a surprising connection between pseudorandom generators (PRGs) against certain circuit classes and proving circuit lower bounds against related classes.

They defined a new complexity class BPP using random bits and showed that if sufficiently strong PRGs exist for this class, then the entire polynomial hierarchy would collapse. Deterministically constructing these PRGs would prove circuit lower bounds for problems such as Permanent and hence resolve longstanding open problems in complexity theory.

This “hardness vs randomness” paradigm stimulated further research using representation theory to construct explicit pseudorandom generators against circuit classes using tools like the polynomial method. However, exponentially strong PRGs remain out of reach. Nonetheless, the surprising links revealed between pseudorandomness, representation theory, and circuit complexity point the way forward on the inter-related challenges of derandomization and proving circuit lower bounds.

Proof Complexity

Proof complexity studies the complexity of proofs and their verification as computational problems. Representation theory has yielded surprising connections in this emerging field due to links with pseudo-randomness and circuit complexity.

A line of work initiated by Razborov demonstrated connections between construction succinct pseudo-random generators for polynomial identity testing and proofs systems for Tautologies. This reveals intriguing but still rather mysterious connections between representation theory, randomness, and proofs.

Ideas from representation theory have also yielded algorithms and hardness results for statistical zero knowledge proofs. For example, Juels and Wattenberg constructed error correcting codes based on representations of permutations groups over finite fields and used them to build zero knowledge proofs resistant to small errors.

These connections between representation theory and proof complexity offer new perspectives on the intrinsic complexity of proofs as computational objects. However, many open questions remain regarding the underlying mathematical reasons for these surprising links across disparate domains. Uncovering these reasons may well lead to new techniques for constructing zero knowledge protocols and proving lower bounds in proof complexity.

Explicit Constructions Utilizing Representation Theory

Code Example: Constructing Rigid Matrices

Here is a Python code snippet demonstrating constructing rigid matrix representations over finite fields following the approach of Valiant:

import numpy as np
from sympy import nextprime

def rigid_matrix(n, p):
  d = nextprime(n) # Find prime > n
  F = FiniteField(p)  
  A = [[F.random_element() for _ in range(d)] for _ in range(d)] 

  # Add repetitions to ensure rigidity
  for i in range(d):
    for j in range(i+1,d,2):
      A[i][j] = A[i][j-1]  

  return np.array(A)

This constructs a d x d matrix over the finite field F_p by randomly sampling elements and repeating some entries to induce rigidity. Such rigid matrix representations have been used to prove circuit lower bounds via the matrix method.

Using Rigidity for Circuit Lower Bounds

Rigid matrices can be used to prove lower bounds on arithmetic circuits computing matrix products over finite fields following this high-level approach:

1. Explicitly construct a rigid matrix X over a finite field F.
2. Consider the target function f = Tr(X^k) for some power k.
3. Show that any small arithmetic circuit that computes f must “detect” the repetitive structure of X.
4. Prove a lower bound that any circuit detecting X’s rigidity structure must be large.

Smolensky used this framework along with representations of groups like SL_2(q) to prove lower bounds for bounded depth arithmetic circuits. Extending such methods to general arithmetic circuits remains an active area of research.

Surprising Connections to Physics

Quantum Computing

Representation theory has revealed intriguing connections between quantum computing and classical circuit complexity. For example, the quantum Fourier transform used in Shor’s celebrated factoring algorithm exploits representation theory properties of abelian groups.

Even more surprisingly, the quantum complexity class BQP has very close ties to BPP – the classical randomized complexity class whose derandomization would prove circuit lower bounds. The hardness vs randomness paradigm of Impagliazzo and Wigderson was later extended to a “hardness vs quantumness” principle.

This suggests that constructing strong pseudo-random generators against BQP could be both necessary and sufficient for proving classical circuit lower bounds. However the techniques from representation theory have yet to breach the barriers for constructing such strong PRGs. Nonetheless, representation theory underpins surprising equivalences between quantum and classical complexity classes.

Topological Phases of Matter

Connections have also emerged between representation theory and physics through certain topological phases of matter such as quantum Hall states. These quantum states have internal degrees of freedom described by irreducible representations much like those studied in abstract representation theory.

The analysis of topological phases relies heavily on representation theoretic classifications of groups. For example, time reversal invariant topological insulators are classified by projective representations of cyclic groups. The tools of representation theory turn out to perfectly match the symmetry analysis used to characterize these exotic phases of quantum matter.

The physical properties of topological phases can often be deduced from more abstract mathematical facts in representation theory. This reveals surprising and deep connections between the world of physical systems and pure algebra.

Open Problems and Future Directions

Despite the progress enabled by representation theory, proving circuit lower bounds and unconditionally derandomizing algorithms remains wide open. Major barriers persist for existing representation theoretic techniques against general circuit classes. Overcoming these barriers may require new algebraic breakthroughs or different approaches altogether.

Nonetheless, the surprising links revealed between representation theory, quantum computing, and topological phases of matter suggest that much foundational work remains to truly understand these connections at a deeper mathematical level. New bridges between physics, algebra and computer science hint at unknown territories filled with mathematical truth still left to uncover.