We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {-1,1}^n. This result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a "regular PTF is a PTF sign(p(x)) where the influence of each variable on the polynomial p(x) is a small fraction of the total influence of p. As an application of this regularity lemma, we prove that for any constants d \geq 1, \eps \geq 0, every degree-d PTF over n variables has can be approximated to accuracy eps by a constant-degree PTF that has integer weights of total magnitude O(n^d). This weight bound is shown to be optimal up to constant factors.We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {-1,1}^n. This result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a "regular PTF is a PTF sign(p(x)) where the influence of each variable on the polynomial p(x) is a small fraction of the total influence of p. As an application of this regularity lemma, we prove that for any constants d \geq 1, \eps \geq 0, every degree-d PTF over n variables has can be approximated to accuracy eps by a constant-degree PTF that has integer weights of total magnitude O(n^d). This weight bound is shown to be optimal up to constant factors.

arXiv:0909.4727Sep 25, 20095 citations

On the Power of Restricted Monte Carlo Algorithms

Stefan Heinrich

MATRIX book seriesJan 1, 20203 citations

Simulated Quantum Annealing Can Be Exponentially Faster Than Classical Simulated Annealing

Elizabeth Crosson, Aram W. Harrow

Can quantum computers solve optimization problems much more quickly than classical computers? One major piece of evidence for this proposition has been the fact that Quantum Annealing (QA) finds the minimum of some cost functions exponentially more quickly than classical Simulated Annealing (SA). One such cost function is the simple “Hamming weight with a spike” function in which the input is an n-bit string and the objective function is simply the Hamming weight, plus a tall thin barrier centered around Hamming weight n/4. While the global minimum of this cost function can be found by inspection, it is also a plausible toy model of the sort of local minima that arise in realworld optimization problems. It was shown by Farhi, Goldstone and Gutmann [1] that for this example SA takes exponential time and QA takes polynomial time, and the same result was generalized by Reichardt [2] to include barriers with width nζ and height nαfor ζ + α ≤ 1/2. This advantage could be explained in terms of quantummechanical “tunneling.” Our work considers a classical algorithm known as Simulated Quantum Annealing (SQA) which relates certain quantum systems to classical Markov chains. By proving that these chains mix rapidly, we show that SQA runs in polynomial time on the Hamming weight with spike problem in much of the parameter regime where QA achieves an exponential advantage over SA. While our analysis only covers this toy model, it can be seen as evidence against the prospect of exponential quantum speedup using tunneling. Our technical contributions include extending the canonical path method for analyzing Markov chains to cover the case when not all vertices can be connected by low-congestion paths. We also develop methods for taking advantage of warm starts and for relating the quantum state in QA to the probability distribution in SQA. These techniques may be of use in future studies of SQA or of rapidly mixing Markov chains in general.Can quantum computers solve optimization problems much more quickly than classical computers? One major piece of evidence for this proposition has been the fact that Quantum Annealing (QA) finds the minimum of some cost functions exponentially more quickly than classical Simulated Annealing (SA). One such cost function is the simple “Hamming weight with a spike” function in which the input is an n-bit string and the objective function is simply the Hamming weight, plus a tall thin barrier centered around Hamming weight n/4. While the global minimum of this cost function can be found by inspection, it is also a plausible toy model of the sort of local minima that arise in realworld optimization problems. It was shown by Farhi, Goldstone and Gutmann [1] that for this example SA takes exponential time and QA takes polynomial time, and the same result was generalized by Reichardt [2] to include barriers with width nζ and height nαfor ζ + α ≤ 1/2. This advantage could be explained in terms of quantummechanical “tunneling.” Our work considers a classical algorithm known as Simulated Quantum Annealing (SQA) which relates certain quantum systems to classical Markov chains. By proving that these chains mix rapidly, we show that SQA runs in polynomial time on the Hamming weight with spike problem in much of the parameter regime where QA achieves an exponential advantage over SA. While our analysis only covers this toy model, it can be seen as evidence against the prospect of exponential quantum speedup using tunneling. Our technical contributions include extending the canonical path method for analyzing Markov chains to cover the case when not all vertices can be connected by low-congestion paths. We also develop methods for taking advantage of warm starts and for relating the quantum state in QA to the probability distribution in SQA. These techniques may be of use in future studies of SQA or of rapidly mixing Markov chains in general.

arXiv:1601.03030Annual Symposium on Foundations of Computer ScienceOct 1, 201676 citations

Benchmark of the Polynomial Expansion Monte Carlo Method

Hiroaki Ishizuka

Springer thesesJan 1, 20150 citations

Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark

Liangdong Lu, Ruipan Yang, Guanmin Guo

Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight- $6$ checks, exemplified by $[[144,12,12]]$ with $kd^2/n=12$ . We develop the algebraic structure theory of BB-type codes with weight- $8$ checks (weight- $4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula $k=2\dim R/(A,B)$ (forcing even $k$ ), a $4\ell m$ -element symmetry group on generator pairs, an $X/Z$ distance equality $d_X=d_Z$ , and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At $n=144$ the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: $[[144,6,d\ge 15]]$ exceeds the benchmark distance $12$ (certified $d\ge 15$ ), $[[144,10,12]]$ reaches it with weight- $8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at $kd^2/n=11.11$ ( $7.4\%$ below benchmark) while decoding no worse. At $n=72$ , $[[72,14,8]]$ attains $kd^2/n=12.44$ ---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight- $8$ codes at $\approx 0.1\%$ pseudo-threshold versus $\approx 0.4\%$ for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-$6$ checks, exemplified by $[[144,12,12]]$ with $kd^2/n=12$. We develop the algebraic structure theory of BB-type codes with weight-$8$ checks (weight-$4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula $k=2\dim R/(A,B)$ (forcing even $k$), a $4\ell m$-element symmetry group on generator pairs, an $X/Z$ distance equality $d_X=d_Z$, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At $n=144$ the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: $[[144,6,d\ge 15]]$ exceeds the benchmark distance $12$ (certified $d\ge 15$), $[[144,10,12]]$ reaches it with weight-$8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at $kd^2/n=11.11$ ($7.4\%$ below benchmark) while decoding no worse. At $n=72$, $[[72,14,8]]$ attains $kd^2/n=12.44$---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-$8$ codes at $\approx 0.1\%$ pseudo-threshold versus $\approx 0.4\%$ for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.

quant-pharXiv:2609.06572Sep 8, 20260 citations

Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-

6

checks, exemplified by

[[144,12,12]]

with

kd^2/n=12

. We develop the algebraic structure theory of BB-type codes with weight-

8

checks (weight-

4

generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula

k=2\dim R/(A,B)

(forcing even

k

), a

4\ell m

-element symmetry group on generator pairs, an

X/Z

distance equality

d_X=d_Z

, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At

n=144

the pipeline returns a census of

53

codes whose strongest members surpass the BB benchmark:

[[144,6,d\ge 15]]

exceeds the benchmark distance

12

(certified

d\ge 15

[[144,10,12]]

reaches it with weight-

8

checks, and

[[144,16,10]]

encodes a third more logical qubits at

kd^2/n=11.11

(

7.4\%

below benchmark) while decoding no worse. At

n=72

[[72,14,8]]

attains

kd^2/n=12.44

---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-

8

codes at

\approx 0.1\%

pseudo-threshold versus

\approx 0.4\%

for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.