Andrei Ruckenstein | Boston University (original) (raw)

Papers by Andrei Ruckenstein

Physical review research, Oct 1, 2019

We construct a model of short-range interacting Ising spins on a translationally invariant two-di... more We construct a model of short-range interacting Ising spins on a translationally invariant two-dimensional lattice that mimics a reversible circuit that multiplies or factorizes integers, depending on the choice of boundary conditions. We prove that, for open boundary conditions, the model exhibits no finite-temperature phase transition. Yet we find that it displays glassy dynamics with astronomically slow relaxation times, numerically consistent with a double exponential dependence on the inverse temperature. The slowness of the dynamics arises due to errors that occur during thermal annealing that cost little energy but flip an extensive number of spins. We argue that the energy barrier that needs to be overcome in order to heal such defects scales linearly with the correlation length, which diverges exponentially with inverse temperature, thus yielding the double exponential behavior of the relaxation time.

Bulletin of the American Physical Society, Mar 9, 2018

Physical Review Letters, Apr 25, 1994

arXiv (Cornell University), Apr 19, 2023

We propose a mechanism for reaching pseudorandom quantum states, computationally indistinguishabl... more We propose a mechanism for reaching pseudorandom quantum states, computationally indistinguishable from Haar random, with shallow log-n depth quantum circuits, where n is the number of qudits. We argue that log n depth 2-qubit-gate-based generic random quantum circuits that are claimed to provide a lower bound on the speed of information scrambling, cannot produce computationally pseudorandom quantum states. This conclusion is connected with the presence of polynomial (in n) tails in the stay probability of short Pauli strings that survive evolution through such shallow circuits. We show, however, that stay-probability-tails can be eliminated and pseudorandom quantum states can be accomplished with shallow log n depth circuits built from a special universal family of 'inflationary' quantum (IQ) gates. We prove that IQ-gates cannot be implemented with 2-qubit gates, but can be realized either as a subset of 2-qudit-gates in U (d 2) with d ≥ 3 and d prime, or as special 3-qubit gates.

Bulletin of the American Physical Society, Mar 9, 2018

Physical review research, Mar 31, 2023

Physical review, Mar 1, 1999

A two-site cluster generalization of the Hubbard model in large dimensions is examined in order t... more A two-site cluster generalization of the Hubbard model in large dimensions is examined in order to study the role of short-range spin correlations near the metal-insulator transition (MIT). The model is mapped to a two-impurity Kondo-Anderson model in a self-consistently determined bath, making it possible to directly address the competition between the Kondo effect and RKKY interactions in a lattice context. Our results indicate that the RKKY interactions lead to qualitative modifications of the MIT scenario even in the absence of long range antiferromagnetic ordering.

Physical review, Jun 1, 2020

We study the level-spacing statistics in the entanglement spectrum of output states of random uni... more We study the level-spacing statistics in the entanglement spectrum of output states of random universal quantum circuits where qubits are subject to a finite probability of projection to the computational basis at each time step. We encounter two phase transitions with increasing projection rate. The first is the volume-to-area law transition observed in quantum circuits with projective measurements. We identify a second transition within the area law phase by repartioning the system randomly into two subsystems and probing the entanglement level statistics. This second transition separates a pure Poisson level statistics phase at large projective measurement rates from a regime of residual level repulsion in the entanglement spectrum, characterized by non-universal level spacing statistics that interpolates between the Wigner-Dyson and Poisson distributions. By applying a tensor network contraction algorithm introduced in Ref. [1] to the circuit spacetime, we identify this second projective-measurement-driven transition as a percolation transition of entangled bonds. The same behavior is observed in both circuits of random two-qubit unitaries and circuits of universal gate sets, including the set implemented by Google in its Sycamore circuits.

Physical review, Jul 1, 1987

We develop the simplest mean-field theory of an extended Hubbard model in the limit of a large in... more We develop the simplest mean-field theory of an extended Hubbard model in the limit of a large intrasite Coulomb interaction, concentrating on the possibility of superconductivity induced by the superexchange interaction and weakened by the intersite Coulomb repulsion. We calculate the critical temperature and the coherence length as a function of filling, as well as the temperature dependence of the magnetic susceptibility and specific heat. Finally, we comment on the physics of the insulating state at half filling, and mention the probable eff'ects of fluctuations.

arXiv (Cornell University), Mar 16, 2022

We introduce a new approach to computation on encrypted data-Encrypted Operator Computing (EOC)-a... more We introduce a new approach to computation on encrypted data-Encrypted Operator Computing (EOC)-as an alternative to Fully Homomorphic Encryption (FHE). Given a plaintext vector | x , x ∈ {0, 1} n , and a function F (x) represented as an operatorF ,F | x = | F (x) , the EOC scheme is based on obfuscating the conjugated operator (circuit)F E =ÊFÊ −1 that implements computation on encrypted data,Ê| x. The construction of EOC hinges on the existence of a two-stage NC 1 reversible-circuit-based IND-CCA2 cipherÊ =NL, whereL andN represent, respectively, linear and non-linear NC 1 tree-structured circuits of 3-bit reversible gates. We make and motivate security assumptions about such a NC 1 cipher. Furthermore, we establish the polynomial complexity of the obfuscated circuit, the evaluator O(F E), by proving that: (a) conjugation of each gate of F withL yields a polynomial number of gates; and (b) the subsequent conjugation withN yields a polynomial number of "chips," n-input/n-output reversible functions, with outputs expressed as polynomial-sized ordered Binary Decision Diagrams (OBDDs). The security of individual chips is connected to the notion of Best Possible Obfuscators [10] which relies on poly-size OBDDs and the fact that OBDDs are normal forms that expose the functionality but hide the gate implementation of the chip. We conjecture that the addition of random pairs of NOTs between layers ofN during the construction of F E , a device analogous to the AddRoundKey rounds of AES, ensures the security of the evaluator. We also present a generalization to asymmetric encryption.

Physical Review Letters, Oct 30, 1989

arXiv (Cornell University), Nov 12, 2020

We cast encryption via classical block ciphers in terms of operator spreading in a dual space of ... more We cast encryption via classical block ciphers in terms of operator spreading in a dual space of Pauli strings, a formulation which allows us to characterize classical ciphers by using tools well known in the analysis of quantum many-body systems. We connect plaintext and ciphertext attacks to out-of-time order correlators (OTOCs) and quantify the quality of ciphers using measures of delocalization in string space such as participation ratios and corresponding entropies obtained from the wave function amplitudes in string space. In particular, we show that in Feistel ciphers the entropy saturates its bound to exponential precision for ciphers with 4 or more rounds, consistent with the classic Luby-Rackoff result that it takes these many rounds to generate strong pseudorandom permutations. The saturation of the string-space information entropy is accompanied by the vanishing of OTOCs. Together these signal irreversibility and chaos, which we take to be the defining properties of good classical ciphers. More precisely, we define a good cipher by requiring that the OTOCs vanish to exponential precision and that the string entropies saturate to the values associated with a random permutation, which are computed explicitly in the paper. In turn, these criteria imply that the cipher cannot be distinguished from a pseudorandom permutation with a polynomial number of queries. We argue that the conditions on both OTOCs and string entropies can be satisfied by n-bit block ciphers implemented via random reversible circuits with O(n log n) gates. This paper focuses on a tree-structured cipher composed of layers of n/3 3-bit gates, for which a "key" specifies uniquely the sequence of gates that comprise the circuit. We show that in order to reach this "speed limit" one must employ a three-stage circuit consisting of a nonlinear stage implemented by layers of nonlinear gates that proliferate the number of strings, flanked by two linear stages, each deploying layers of a special set of linear "inflationary" gates that accelerate the growth of small individual strings. The close formal correspondence to quantum scramblers established in this work leads us to suggest that this three-stage construction is also required in order to scramble quantum states to similar precision and with circuits of similar size. A shallow, O(log n)-depth cipher of the type described here can be used in constructing a polynomial-overhead scheme for computation on encrypted data proposed in another publication as an alternative to Homomorphic Encryption.

Bulletin of the American Physical Society, Mar 9, 2018

Bulletin of the American Physical Society, Mar 8, 2018

arXiv (Cornell University), Aug 29, 2017

We develop a tensor network technique that can solve universal reversible classical computational... more We develop a tensor network technique that can solve universal reversible classical computational problems, formulated as vertex models on a square lattice [Nat. Commun. 8, 15303 (2017)]. By encoding the truth table of each vertex constraint in a tensor, the total number of solutions compatible with partial inputs/outputs at the boundary can be represented as the full contraction of a tensor network. We introduce an iterative compression-decimation (ICD) scheme that performs this contraction efficiently. The ICD algorithm first propagates local constraints to longer ranges via repeated contraction-decomposition sweeps over all lattice bonds, thus achieving compression on a given length scale. It then decimates the lattice via coarse-graining tensor contractions. Repeated iterations of these two steps gradually collapse the tensor network and ultimately yield the exact tensor trace for large systems, without the need for manual control of tensor dimensions. Our protocol allows us to obtain the exact number of solutions for computations where a naive enumeration would take astronomically long times.

arXiv (Cornell University), Apr 18, 2016

We construct a planar vertex model that encodes the result of a universal reversible classical co... more We construct a planar vertex model that encodes the result of a universal reversible classical computation in its ground state. The approach involves Boolean variables (spins) placed on links of a two-dimensional lattice, with vertices representing logic gates. Large short-ranged interactions between at most two spins implement the operation of each gate. The lattice is anisotropic with one direction corresponding to "computational" time, and with transverse boundaries storing the computation's input and output. Our approach tackles both fixed input computations that proceed forward in computational time, but also, more interestingly, problems in which only partial information about both inputs and outputs is known. In that case, reaching the ground state requires flow of information both forwards and backwards across the lattice, processes that are naturally built into our mapping of reversible computations into the vertex model. This allows us to tackle a subclass of the Circuit Satisfiability (CSAT) problem and to solve factoring problems by using multiplication circuits with polynomial depth. While we show that the model displays no finite temperature phase transitions, independent of circuit, the computational complexity is encoded in the scaling of the relaxation rate into the ground state with the system size. To explore faster relaxation routes, we construct an explicit mapping of the vertex model into the Chimera architecture of the D-Wave machine, initiating a novel approach to reversible classical computation based on state-of-the-art implementations of quantum annealing.

arXiv (Cornell University), Dec 7, 2022

arXiv (Cornell University), Nov 12, 2020

Physical review research, Oct 1, 2019

We construct a model of short-range interacting Ising spins on a translationally invariant two-di... more We construct a model of short-range interacting Ising spins on a translationally invariant two-dimensional lattice that mimics a reversible circuit that multiplies or factorizes integers, depending on the choice of boundary conditions. We prove that, for open boundary conditions, the model exhibits no finite-temperature phase transition. Yet we find that it displays glassy dynamics with astronomically slow relaxation times, numerically consistent with a double exponential dependence on the inverse temperature. The slowness of the dynamics arises due to errors that occur during thermal annealing that cost little energy but flip an extensive number of spins. We argue that the energy barrier that needs to be overcome in order to heal such defects scales linearly with the correlation length, which diverges exponentially with inverse temperature, thus yielding the double exponential behavior of the relaxation time.

Bulletin of the American Physical Society, Mar 9, 2018

Physical Review Letters, Apr 25, 1994

arXiv (Cornell University), Apr 19, 2023

We propose a mechanism for reaching pseudorandom quantum states, computationally indistinguishabl... more We propose a mechanism for reaching pseudorandom quantum states, computationally indistinguishable from Haar random, with shallow log-n depth quantum circuits, where n is the number of qudits. We argue that log n depth 2-qubit-gate-based generic random quantum circuits that are claimed to provide a lower bound on the speed of information scrambling, cannot produce computationally pseudorandom quantum states. This conclusion is connected with the presence of polynomial (in n) tails in the stay probability of short Pauli strings that survive evolution through such shallow circuits. We show, however, that stay-probability-tails can be eliminated and pseudorandom quantum states can be accomplished with shallow log n depth circuits built from a special universal family of 'inflationary' quantum (IQ) gates. We prove that IQ-gates cannot be implemented with 2-qubit gates, but can be realized either as a subset of 2-qudit-gates in U (d 2) with d ≥ 3 and d prime, or as special 3-qubit gates.

Bulletin of the American Physical Society, Mar 9, 2018

Physical review research, Mar 31, 2023

Physical review, Mar 1, 1999

A two-site cluster generalization of the Hubbard model in large dimensions is examined in order t... more A two-site cluster generalization of the Hubbard model in large dimensions is examined in order to study the role of short-range spin correlations near the metal-insulator transition (MIT). The model is mapped to a two-impurity Kondo-Anderson model in a self-consistently determined bath, making it possible to directly address the competition between the Kondo effect and RKKY interactions in a lattice context. Our results indicate that the RKKY interactions lead to qualitative modifications of the MIT scenario even in the absence of long range antiferromagnetic ordering.

Physical review, Jun 1, 2020

We study the level-spacing statistics in the entanglement spectrum of output states of random uni... more We study the level-spacing statistics in the entanglement spectrum of output states of random universal quantum circuits where qubits are subject to a finite probability of projection to the computational basis at each time step. We encounter two phase transitions with increasing projection rate. The first is the volume-to-area law transition observed in quantum circuits with projective measurements. We identify a second transition within the area law phase by repartioning the system randomly into two subsystems and probing the entanglement level statistics. This second transition separates a pure Poisson level statistics phase at large projective measurement rates from a regime of residual level repulsion in the entanglement spectrum, characterized by non-universal level spacing statistics that interpolates between the Wigner-Dyson and Poisson distributions. By applying a tensor network contraction algorithm introduced in Ref. [1] to the circuit spacetime, we identify this second projective-measurement-driven transition as a percolation transition of entangled bonds. The same behavior is observed in both circuits of random two-qubit unitaries and circuits of universal gate sets, including the set implemented by Google in its Sycamore circuits.

Physical review, Jul 1, 1987

We develop the simplest mean-field theory of an extended Hubbard model in the limit of a large in... more We develop the simplest mean-field theory of an extended Hubbard model in the limit of a large intrasite Coulomb interaction, concentrating on the possibility of superconductivity induced by the superexchange interaction and weakened by the intersite Coulomb repulsion. We calculate the critical temperature and the coherence length as a function of filling, as well as the temperature dependence of the magnetic susceptibility and specific heat. Finally, we comment on the physics of the insulating state at half filling, and mention the probable eff'ects of fluctuations.

arXiv (Cornell University), Mar 16, 2022

We introduce a new approach to computation on encrypted data-Encrypted Operator Computing (EOC)-a... more We introduce a new approach to computation on encrypted data-Encrypted Operator Computing (EOC)-as an alternative to Fully Homomorphic Encryption (FHE). Given a plaintext vector | x , x ∈ {0, 1} n , and a function F (x) represented as an operatorF ,F | x = | F (x) , the EOC scheme is based on obfuscating the conjugated operator (circuit)F E =ÊFÊ −1 that implements computation on encrypted data,Ê| x. The construction of EOC hinges on the existence of a two-stage NC 1 reversible-circuit-based IND-CCA2 cipherÊ =NL, whereL andN represent, respectively, linear and non-linear NC 1 tree-structured circuits of 3-bit reversible gates. We make and motivate security assumptions about such a NC 1 cipher. Furthermore, we establish the polynomial complexity of the obfuscated circuit, the evaluator O(F E), by proving that: (a) conjugation of each gate of F withL yields a polynomial number of gates; and (b) the subsequent conjugation withN yields a polynomial number of "chips," n-input/n-output reversible functions, with outputs expressed as polynomial-sized ordered Binary Decision Diagrams (OBDDs). The security of individual chips is connected to the notion of Best Possible Obfuscators [10] which relies on poly-size OBDDs and the fact that OBDDs are normal forms that expose the functionality but hide the gate implementation of the chip. We conjecture that the addition of random pairs of NOTs between layers ofN during the construction of F E , a device analogous to the AddRoundKey rounds of AES, ensures the security of the evaluator. We also present a generalization to asymmetric encryption.

Physical Review Letters, Oct 30, 1989

arXiv (Cornell University), Nov 12, 2020

We cast encryption via classical block ciphers in terms of operator spreading in a dual space of ... more We cast encryption via classical block ciphers in terms of operator spreading in a dual space of Pauli strings, a formulation which allows us to characterize classical ciphers by using tools well known in the analysis of quantum many-body systems. We connect plaintext and ciphertext attacks to out-of-time order correlators (OTOCs) and quantify the quality of ciphers using measures of delocalization in string space such as participation ratios and corresponding entropies obtained from the wave function amplitudes in string space. In particular, we show that in Feistel ciphers the entropy saturates its bound to exponential precision for ciphers with 4 or more rounds, consistent with the classic Luby-Rackoff result that it takes these many rounds to generate strong pseudorandom permutations. The saturation of the string-space information entropy is accompanied by the vanishing of OTOCs. Together these signal irreversibility and chaos, which we take to be the defining properties of good classical ciphers. More precisely, we define a good cipher by requiring that the OTOCs vanish to exponential precision and that the string entropies saturate to the values associated with a random permutation, which are computed explicitly in the paper. In turn, these criteria imply that the cipher cannot be distinguished from a pseudorandom permutation with a polynomial number of queries. We argue that the conditions on both OTOCs and string entropies can be satisfied by n-bit block ciphers implemented via random reversible circuits with O(n log n) gates. This paper focuses on a tree-structured cipher composed of layers of n/3 3-bit gates, for which a "key" specifies uniquely the sequence of gates that comprise the circuit. We show that in order to reach this "speed limit" one must employ a three-stage circuit consisting of a nonlinear stage implemented by layers of nonlinear gates that proliferate the number of strings, flanked by two linear stages, each deploying layers of a special set of linear "inflationary" gates that accelerate the growth of small individual strings. The close formal correspondence to quantum scramblers established in this work leads us to suggest that this three-stage construction is also required in order to scramble quantum states to similar precision and with circuits of similar size. A shallow, O(log n)-depth cipher of the type described here can be used in constructing a polynomial-overhead scheme for computation on encrypted data proposed in another publication as an alternative to Homomorphic Encryption.

Bulletin of the American Physical Society, Mar 9, 2018

Bulletin of the American Physical Society, Mar 8, 2018

arXiv (Cornell University), Aug 29, 2017

We develop a tensor network technique that can solve universal reversible classical computational... more We develop a tensor network technique that can solve universal reversible classical computational problems, formulated as vertex models on a square lattice [Nat. Commun. 8, 15303 (2017)]. By encoding the truth table of each vertex constraint in a tensor, the total number of solutions compatible with partial inputs/outputs at the boundary can be represented as the full contraction of a tensor network. We introduce an iterative compression-decimation (ICD) scheme that performs this contraction efficiently. The ICD algorithm first propagates local constraints to longer ranges via repeated contraction-decomposition sweeps over all lattice bonds, thus achieving compression on a given length scale. It then decimates the lattice via coarse-graining tensor contractions. Repeated iterations of these two steps gradually collapse the tensor network and ultimately yield the exact tensor trace for large systems, without the need for manual control of tensor dimensions. Our protocol allows us to obtain the exact number of solutions for computations where a naive enumeration would take astronomically long times.

arXiv (Cornell University), Apr 18, 2016

We construct a planar vertex model that encodes the result of a universal reversible classical co... more We construct a planar vertex model that encodes the result of a universal reversible classical computation in its ground state. The approach involves Boolean variables (spins) placed on links of a two-dimensional lattice, with vertices representing logic gates. Large short-ranged interactions between at most two spins implement the operation of each gate. The lattice is anisotropic with one direction corresponding to "computational" time, and with transverse boundaries storing the computation's input and output. Our approach tackles both fixed input computations that proceed forward in computational time, but also, more interestingly, problems in which only partial information about both inputs and outputs is known. In that case, reaching the ground state requires flow of information both forwards and backwards across the lattice, processes that are naturally built into our mapping of reversible computations into the vertex model. This allows us to tackle a subclass of the Circuit Satisfiability (CSAT) problem and to solve factoring problems by using multiplication circuits with polynomial depth. While we show that the model displays no finite temperature phase transitions, independent of circuit, the computational complexity is encoded in the scaling of the relaxation rate into the ground state with the system size. To explore faster relaxation routes, we construct an explicit mapping of the vertex model into the Chimera architecture of the D-Wave machine, initiating a novel approach to reversible classical computation based on state-of-the-art implementations of quantum annealing.

arXiv (Cornell University), Dec 7, 2022

arXiv (Cornell University), Nov 12, 2020