19 research outputs found
Good approximate quantum LDPC codes from spacetime circuit Hamiltonians
We study approximate quantum low-density parity-check (QLDPC) codes, which
are approximate quantum error-correcting codes specified as the ground space of
a frustration-free local Hamiltonian, whose terms do not necessarily commute.
Such codes generalize stabilizer QLDPC codes, which are exact quantum
error-correcting codes with sparse, low-weight stabilizer generators (i.e. each
stabilizer generator acts on a few qubits, and each qubit participates in a few
stabilizer generators). Our investigation is motivated by an important question
in Hamiltonian complexity and quantum coding theory: do stabilizer QLDPC codes
with constant rate, linear distance, and constant-weight stabilizers exist?
We show that obtaining such optimal scaling of parameters (modulo
polylogarithmic corrections) is possible if we go beyond stabilizer codes: we
prove the existence of a family of approximate QLDPC
codes that encode logical qubits into physical
qubits with distance and approximation infidelity
. The code space is
stabilized by a set of 10-local noncommuting projectors, with each physical
qubit only participating in projectors. We
prove the existence of an efficient encoding map, and we show that arbitrary
Pauli errors can be locally detected by circuits of polylogarithmic depth.
Finally, we show that the spectral gap of the code Hamiltonian is
by analyzing a spacetime circuit-to-Hamiltonian
construction for a bitonic sorting network architecture that is spatially local
in dimensions.Comment: 51 pages, 13 figure
Metastability-containing circuits, parallel distance problems, and terrain guarding
We study three problems. The first is the phenomenon of metastability in digital circuits. This is a state of bistable storage elements, such as registers, that is neither logical 0 nor 1 and breaks the abstraction of Boolean logic. We propose a time- and value-discrete model for metastability in digital circuits and show that it reflects relevant physical properties. Further, we propose the fundamentally new approach of using logical masking to perform meaningful computations despite the presence of metastable upsets and analyze what functions can be computed in our model. Additionally, we show that circuits with masking registers grow computationally more powerful with each available clock cycle. The second topic are parallel algorithms, based on an algebraic abstraction of the Moore-Bellman-Ford algorithm, for solving various distance problems. Our focus are distance approximations that obey the triangle inequality while at the same time achieving polylogarithmic depth and low work. Finally, we study the continuous Terrain Guarding Problem. We show that it has a rational discretization with a quadratic number of guard candidates, establish its membership in NP and the existence of a PTAS, and present an efficient implementation of a solver.Wir betrachten drei Probleme, zunรคchst das Phรคnomen von Metastabilitรคt in digitalen Schaltungen. Dabei geht es um einen Zustand in bistabilen Speicherelementen, z.B. Registern, welcher weder logisch 0 noch 1 entspricht und die Abstraktion Boolescher Logik unterwandert. Wir prรคsentieren ein zeit- und wertdiskretes Modell fรผr Metastabilitรคt in digitalen Schaltungen und zeigen, dass es relevante physikalische Eigenschaften abbildet. Des Weiteren prรคsentieren wir den grundlegend neuen Ansatz, trotz auftretender Metastabilitรคt mit Hilfe von logischem Maskieren sinnvolle Berechnungen durchzufรผhren und bestimmen, welche Funktionen in unserem Modell berechenbar sind. Darรผber hinaus zeigen wir, dass durch Maskingregister in zusรคtzlichen Taktzyklen mehr Funktionen berechenbar werden. Das zweite Thema sind parallele Algorithmen die, basierend auf einer Algebraisierung des Moore-Bellman-Ford-Algorithmus, diverse Distanzprobleme lรถsen. Der Fokus liegt auf Distanzapproximationen unter Einhaltung der Dreiecksungleichung bei polylogarithmischer Tiefe und niedriger Arbeit. Abschlieรend betrachten wir das kontinuierliche Terrain Guarding Problem. Wir zeigen, dass es eine rationale Diskretisierung mit einer quadratischen Anzahl von Wรคchterpositionen erlaubt, folgern dass es in NP liegt und ein PTAS existiert und prรคsentieren eine effiziente Implementierung, die es lรถst
Instantaneous Non-Local Computation of Low T-Depth Quantum Circuits
Instantaneous non-local quantum computation requires multiple parties to jointly perform a quantum operation, using pre-shared entanglement and a single round of simultaneous communication. We study this task for its close connection to position-based quantum cryptography, but it also has natural applications in the context of foundations of quantum physics and in distributed computing. The best known general construction for instantaneous non-local quantum computation requires a pre-shared state which is exponentially large in the number of qubits involved in the operation, while efficient constructions are known for very specific cases only.
We partially close this gap by presenting new schemes for efficient instantaneous non-local computation of several classes of quantum circuits, using the Clifford+T gate set. Our main result is a protocol which uses entanglement exponential in the T-depth of a quantum circuit, able to perform non-local computation of quantum circuits with a (poly-)logarithmic number of layers of T gates with quasi-polynomial entanglement. Our proofs combine ideas from blind and delegated quantum computation with the garden-hose model, a combinatorial model of communication complexity which was recently introduced as a tool for studying certain schemes for quantum position verification. As an application of our results, we also present an efficient attack on a recently-proposed scheme for position verification by Chakraborty and Leverrier
Compressibility-Aware Quantum Algorithms on Strings
Sublinear time quantum algorithms have been established for many fundamental
problems on strings. This work demonstrates that new, faster quantum algorithms
can be designed when the string is highly compressible. We focus on two popular
and theoretically significant compression algorithms -- the Lempel-Ziv77
algorithm (LZ77) and the Run-length-encoded Burrows-Wheeler Transform (RL-BWT),
and obtain the results below.
We first provide a quantum algorithm running in time
for finding the LZ77 factorization of an input string with
factors. Combined with multiple existing results, this yields an
time quantum algorithm for finding the RL-BWT encoding
with BWT runs. Note that . We complement these
results with lower bounds proving that our algorithms are optimal (up to
polylog factors).
Next, we study the problem of compressed indexing, where we provide a
time quantum algorithm for constructing a recently
designed space structure with equivalent capabilities as the
suffix tree. This data structure is then applied to numerous problems to obtain
sublinear time quantum algorithms when the input is highly compressible. For
example, we show that the longest common substring of two strings of total
length can be computed in time, where is the
number of factors in the LZ77 factorization of their concatenation. This beats
the best known time quantum algorithm when is
sufficiently small
๊ทผ์ฌ ์ฐ์ฐ์ ๋ํ ๊ณ์ฐ ๊ฒ์ฆ ์ฐ๊ตฌ
ํ์๋
ผ๋ฌธ(๋ฐ์ฌ)--์์ธ๋ํ๊ต ๋ํ์ :์์ฐ๊ณผํ๋ํ ์๋ฆฌ๊ณผํ๋ถ,2020. 2. ์ฒ์ ํฌ.Verifiable Computing (VC) is a complexity-theoretic method to secure the integrity of computations. The need is increasing as more computations are outsourced to untrusted parties, e.g., cloud platforms. Existing techniques, however, have mainly focused on exact computations, but not approximate arithmetic, e.g., floating-point or fixed-point arithmetic. This makes it hard to apply them to certain types of computations (e.g., machine learning, data analysis, and scientific computation) that inherently require approximate arithmetic.
In this thesis, we present an efficient interactive proof system for arithmetic circuits with rounding gates that can represent approximate arithmetic. The main idea is to represent the rounding gate into a small sub-circuit, and reuse the machinery of the Goldwasser, Kalai, and Rothblum's protocol (also known as the GKR protocol) and its recent refinements. Specifically, we shift the algebraic structure from a field to a ring to better deal with the notion of ``digits'', and generalize the original GKR protocol over a ring. Then, we represent the rounding operation by a low-degree polynomial over a ring, and develop a novel, optimal circuit construction of an arbitrary polynomial to transform the rounding polynomial to an optimal circuit representation. Moreover, we further optimize the proof generation cost for rounding by employing a Galois ring. We provide experimental results that show the efficiency of our system for approximate arithmetic. For example, our implementation performed two orders of magnitude better than the existing system for a nested 128 x 128 matrix multiplication of depth 12 on the 16-bit fixed-point arithmetic.๊ณ์ฐ๊ฒ์ฆ ๊ธฐ์ ์ ๊ณ์ฐ์ ๋ฌด๊ฒฐ์ฑ์ ํ๋ณดํ๊ธฐ ์ํ ๊ณ์ฐ ๋ณต์ก๋ ์ด๋ก ์ ๋ฐฉ๋ฒ์ด๋ค. ์ต๊ทผ ๋ง์ ๊ณ์ฐ์ด ํด๋ผ์ฐ๋ ํ๋ซํผ๊ณผ ๊ฐ์ ์ 3์์๊ฒ ์ธ์ฃผ๋จ์ ๋ฐ๋ผ ๊ทธ ํ์์ฑ์ด ์ฆ๊ฐํ๊ณ ์๋ค. ๊ทธ๋ฌ๋ ๊ธฐ์กด์ ๊ณ์ฐ๊ฒ์ฆ ๊ธฐ์ ์ ๋น๊ทผ์ฌ ์ฐ์ฐ๋ง์ ๊ณ ๋ คํ์ ๋ฟ, ๊ทผ์ฌ ์ฐ์ฐ (๋ถ๋ ์์์ ๋๋ ๊ณ ์ ์์์ ์ฐ์ฐ)์ ๊ณ ๋ คํ์ง ์์๋ค. ๋ฐ๋ผ์ ๋ณธ์ง์ ์ผ๋ก ๊ทผ์ฌ ์ฐ์ฐ์ด ํ์ํ ํน์ ์ ํ์ ๊ณ์ฐ (๊ธฐ๊ณ ํ์ต, ๋ฐ์ดํฐ ๋ถ์ ๋ฐ ๊ณผํ ๊ณ์ฐ ๋ฑ)์ ์ ์ฉํ๊ธฐ ์ด๋ ต๋ค๋ ๋ฌธ์ ๊ฐ ์์๋ค.
์ด ๋
ผ๋ฌธ์ ๋ฐ์ฌ๋ฆผ ๊ฒ์ดํธ๋ฅผ ์๋ฐํ๋ ์ฐ์ ํ๋ก๋ฅผ ์ํ ํจ์จ์ ์ธ ๋ํํ ์ฆ๋ช
์์คํ
์ ์ ์ํ๋ค. ์ด๋ฌํ ์ฐ์ ํ๋ก๋ ๊ทผ์ฌ ์ฐ์ฐ์ ํจ์จ์ ์ผ๋ก ํํํ ์ ์์ผ๋ฏ๋ก, ๊ทผ์ฌ ์ฐ์ฐ์ ๋ํ ํจ์จ์ ์ธ ๊ณ์ฐ ๊ฒ์ฆ์ด ๊ฐ๋ฅํ๋ค. ์ฃผ์ ์์ด๋์ด๋ ๋ฐ์ฌ๋ฆผ ๊ฒ์ดํธ๋ฅผ ์์ ํ๋ก๋ก ๋ณํํ ํ, ์ฌ๊ธฐ์ Goldwasser, Kalai, ๋ฐ Rothblum์ ํ๋กํ ์ฝ (GKR ํ๋กํ ์ฝ)๊ณผ ์ต๊ทผ์ ๊ฐ์ ์ ์ ์ฉํ๋ ๊ฒ์ด๋ค. ๊ตฌ์ฒด์ ์ผ๋ก, ๋์์ ๊ฐ์ฒด๋ฅผ ์ ํ์ฒด๊ฐ ์๋ ``์ซ์''๋ฅผ ๋ณด๋ค ์ ์ฒ๋ฆฌํ ์ ์๋ ํ์ผ๋ก ์นํํ ํ, ํ ์์์ ์ ์ฉ ๊ฐ๋ฅํ๋๋ก ๊ธฐ์กด์ GKR ํ๋กํ ์ฝ์ ์ผ๋ฐํํ์๋ค. ์ดํ, ๋ฐ์ฌ๋ฆผ ์ฐ์ฐ์ ํ์์ ์ฐจ์๊ฐ ๋ฎ์ ๋คํญ์์ผ๋ก ํํํ๊ณ , ๋คํญ์ ์ฐ์ฐ์ ์ต์ ์ ํ๋ก ํํ์ผ๋ก ๋ํ๋ด๋ ์๋กญ๊ณ ์ต์ ํ๋ ํ๋ก ๊ตฌ์ฑ์ ๊ฐ๋ฐํ์๋ค. ๋ํ, ๊ฐ๋ฃจ์ ํ์ ์ฌ์ฉํ์ฌ ๋ฐ์ฌ๋ฆผ์ ์ํ ์ฆ๋ช
์์ฑ ๋น์ฉ์ ๋์ฑ ์ต์ ํํ์๋ค. ๋ง์ง๋ง์ผ๋ก, ์คํ์ ํตํด ์ฐ๋ฆฌ์ ๊ทผ์ฌ ์ฐ์ฐ ๊ฒ์ฆ ์์คํ
์ ํจ์จ์ฑ์ ํ์ธํ์๋ค. ์๋ฅผ ๋ค์ด, ์ฐ๋ฆฌ์ ์์คํ
์ ๊ตฌํ ์, 16 ๋นํธ ๊ณ ์ ์์์ ์ฐ์ฐ์ ํตํ ๊น์ด 12์ ๋ฐ๋ณต๋ 128 x 128 ํ๋ ฌ ๊ณฑ์
์ ๊ฒ์ฆ์ ์์ด ๊ธฐ์กด ์์คํ
๋ณด๋ค ์ฝ 100๋ฐฐ ๋ ๋์ ์ฑ๋ฅ์ ๋ณด์ธ๋ค.1 Introduction 1
1.1 Verifiable Computing 2
1.2 Verifiable Approximate Arithmetic 3
1.2.1 Problem: Verification of Rounding Arithmetic 3
1.2.2 Motivation: Verifiable Machine Learning (AI) 4
1.3 List of Papers 5
2 Preliminaries 6
2.1 Interactive Proof and Argument 6
2.2 Sum-Check Protocol 7
2.3 The GKR Protocol 10
2.4 Notation and Cost Model 14
3 Related Work 15
3.1 Interactive Proofs 15
3.2 (Non-)Interactive Arguments 17
4 Interactive Proof for Rounding Arithmetic 20
4.1 Overview of Our Approach and Result 20
4.2 Interactive Proof over a Ring 26
4.2.1 Sum-Check Protocol over a Ring 27
4.2.2 The GKR Protocol over a Ring 29
4.3 Verifiable Rounding Operation 31
4.3.1 Lowest-Digit-Removal Polynomial over Z_{p^e} 32
4.3.2 Verification of Division-by-p Layer 33
4.4 Delegation of Polynomial Evaluation in Optimal Cost 34
4.4.1 Overview of Our Circuit Construction 35
4.4.2 Our Circuit for Polynomial Evaluation 37
4.4.3 Cost Analysis 40
4.5 Cost Optimization 45
4.5.1 Galois Ring over Z_{p^e} and a Sampling Set 45
4.5.2 Optimization of Prover's Cost for Rounding Layers 47
5 Experimental Results 50
5.1 Experimental Setup 50
5.2 Verifiable Rounding Operation 51
5.2.1 Effectiveness of Optimization via Galois Ring 51
5.2.2 Efficiency of Verifiable Rounding Operation 53
5.3 Comparison to Thaler's Refinement of GKR Protocol 54
5.4 Discussion 57
6 Conclusions 60
6.1 Towards Verifiable AI 61
6.2 Verifiable Cryptographic Computation 62
Abstract (in Korean) 74Docto
Efficient Quantum State Synthesis with One Query
We present a polynomial-time quantum algorithm making a single query (in
superposition) to a classical oracle, such that for every state
there exists a choice of oracle that makes the algorithm construct an
exponentially close approximation of . Previous algorithms for
this problem either used a linear number of queries and polynomial time
[arXiv:1607.05256], or a constant number of queries and polynomially many
ancillae but no nontrivial bound on the runtime [arXiv:2111.02999]. As
corollaries we do the following:
- We simplify the proof that statePSPACE stateQIP
[arXiv:2108.07192] (a quantum state analogue of PSPACE IP) and show
that a constant number of rounds of interaction suffices.
- We show that QAC lower bounds for constructing explicit
states would imply breakthrough circuit lower bounds for computing explicit
boolean functions.
- We prove that every -qubit state can be constructed to within 0.01 error
by an -size circuit over an appropriate finite gate set. More
generally we give a size-error tradeoff which, by a counting argument, is
optimal for any finite gate set.Comment: 39 page
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
The Computational Power of Non-interacting Particles
Shortened abstract: In this thesis, I study two restricted models of quantum
computing related to free identical particles.
Free fermions correspond to a set of two-qubit gates known as matchgates.
Matchgates are classically simulable when acting on nearest neighbors on a
path, but universal for quantum computing when acting on distant qubits or when
SWAP gates are available. I generalize these results in two ways. First, I show
that SWAP is only one in a large family of gates that uplift matchgates to
quantum universality. In fact, I show that the set of all matchgates plus any
nonmatchgate parity-preserving two-qubit gate is universal, and interpret this
fact in terms of local invariants of two-qubit gates. Second, I investigate the
power of matchgates in arbitrary connectivity graphs, showing they are
universal on any connected graph other than a path or a cycle, and classically
simulable on a cycle. I also prove the same dichotomy for the XY interaction.
Free bosons give rise to a model known as BosonSampling. BosonSampling
consists of (i) preparing a Fock state of n photons, (ii) interfering these
photons in an m-mode linear interferometer, and (iii) measuring the output in
the Fock basis. Sampling approximately from the resulting distribution should
be classically hard, under reasonable complexity assumptions. Here I show that
exact BosonSampling remains hard even if the linear-optical circuit has
constant depth. I also report several experiments where three-photon
interference was observed in integrated interferometers of various sizes,
providing some of the first implementations of BosonSampling in this regime.
The experiments also focus on the bosonic bunching behavior and on validation
of BosonSampling devices. This thesis contains descriptions of the numerical
analyses done on the experimental data, omitted from the corresponding
publications.Comment: PhD Thesis, defended at Universidade Federal Fluminense on March
2014. Final version, 208 pages. New results in Chapter 5 correspond to
arXiv:1106.1863, arXiv:1207.2126, and arXiv:1308.1463. New results in Chapter
6 correspond to arXiv:1212.2783, arXiv:1305.3188, arXiv:1311.1622 and
arXiv:1412.678