Coauthors Adam Bene Watts Robin Kothari Avishay Tal January 30, - PowerPoint PPT Presentation

Parity Halving Problem Parity Halving Game Nonlocal n player game: each player gets one input bit x j , responsible for one output bit y j . The players win if | y | ≡ 1 2 | x | (mod 2) Special case n = 3 is the GHZ game. January 30, 2019 11 / 51

Parity Halving Problem Parity Halving Game Nonlocal n player game: each player gets one input bit x j , responsible for one output bit y j . The players win if | y | ≡ 1 2 | x | (mod 2) Special case n = 3 is the GHZ game. General case independently discovered by Mermin (1990) and Brassard, Broadbent, Tapp (2005). January 30, 2019 11 / 51

Parity Halving Problem Quantum Strategy PHP: Given even parity x , find y such that | y | ≡ 1 2 | x | (mod 2). Theorem 1 Given the state | � = 2 ( | 0 · · · 0 � + | 1 · · · 1 � ) , quantum players can always win. √ January 30, 2019 12 / 51

Parity Halving Problem Quantum Strategy PHP: Given even parity x , find y such that | y | ≡ 1 2 | x | (mod 2). Theorem 1 Given the state | � = 2 ( | 0 · · · 0 � + | 1 · · · 1 � ) , quantum players can always win. √ Proof. 0 i ) to their qubit if x j = 1. State is | 0 · · · 0 � + i | x | | 1 · · · 1 � . Each player applies S = ( 1 0 January 30, 2019 12 / 51

Parity Halving Problem Quantum Strategy PHP: Given even parity x , find y such that | y | ≡ 1 2 | x | (mod 2). Theorem 1 Given the state | � = 2 ( | 0 · · · 0 � + | 1 · · · 1 � ) , quantum players can always win. √ Proof. 0 i ) to their qubit if x j = 1. State is | 0 · · · 0 � + i | x | | 1 · · · 1 � . Each player applies S = ( 1 0 | 0 · · · 0 � + | 1 · · · 1 � | 0 · · · 0 � − | 1 · · · 1 � January 30, 2019 12 / 51

Parity Halving Problem Quantum Strategy PHP: Given even parity x , find y such that | y | ≡ 1 2 | x | (mod 2). Theorem 1 Given the state | � = 2 ( | 0 · · · 0 � + | 1 · · · 1 � ) , quantum players can always win. √ Proof. 0 i ) to their qubit if x j = 1. State is | 0 · · · 0 � + i | x | | 1 · · · 1 � . Each player applies S = ( 1 0 | 0 · · · 0 � + | 1 · · · 1 � H ⊗ n | 0 · · · 0 � − | 1 · · · 1 � H ⊗ n � � − − → | x � − − → | x � | x | even | x | odd January 30, 2019 12 / 51

Parity Halving Problem Quantum Strategy PHP: Given even parity x , find y such that | y | ≡ 1 2 | x | (mod 2). Theorem 1 Given the state | � = 2 ( | 0 · · · 0 � + | 1 · · · 1 � ) , quantum players can always win. √ Proof. 0 i ) to their qubit if x j = 1. State is | 0 · · · 0 � + i | x | | 1 · · · 1 � . Each player applies S = ( 1 0 | 0 · · · 0 � + | 1 · · · 1 � H ⊗ n | 0 · · · 0 � − | 1 · · · 1 � H ⊗ n � � − − → | x � − − → | x � | x | even | x | odd Therefore all players apply a Hadamard, measure, and output the result. January 30, 2019 12 / 51

Parity Halving Problem QNC 0 / qpoly circuit x 1 x 1 x 2 x 2 x 3 x 3 y 1 S H y 2 | � S H y 3 S H January 30, 2019 13 / 51

Parity Halving Problem Classical Strategy PHP: Given even parity x , find y such that | y | = 1 2 | x | (mod 2). Theorem (Game Hardness – Brassard, Broadbent, Tapp) Any deterministic strategy wins on a random input with probability at most 1 2 + 2 −⌈ n / 2 ⌉ . January 30, 2019 14 / 51

Parity Halving Problem Classical Strategy PHP: Given even parity x , find y such that | y | = 1 2 | x | (mod 2). Theorem (Game Hardness – Brassard, Broadbent, Tapp) Any deterministic strategy wins on a random input with probability at most 1 2 + 2 −⌈ n / 2 ⌉ . Vague Intuition Output parity depends on input HW modulo 4. January 30, 2019 14 / 51

Parity Halving Problem Classical Strategy PHP: Given even parity x , find y such that | y | = 1 2 | x | (mod 2). Theorem (Game Hardness – Brassard, Broadbent, Tapp) Any deterministic strategy wins on a random input with probability at most 1 2 + 2 −⌈ n / 2 ⌉ . Vague Intuition Output parity depends on input HW modulo 4. Any one bit is almost completely independent of HW mod 4. January 30, 2019 14 / 51

Parity Halving Problem Classical Strategy PHP: Given even parity x , find y such that | y | = 1 2 | x | (mod 2). Theorem (Game Hardness – Brassard, Broadbent, Tapp) Any deterministic strategy wins on a random input with probability at most 1 2 + 2 −⌈ n / 2 ⌉ . Vague Intuition Output parity depends on input HW modulo 4. Any one bit is almost completely independent of HW mod 4. Fraction of strings with HW i (mod 4) is 1 4 + O (2 − n / 2 ). January 30, 2019 14 / 51

Parity Halving Problem Locality in circuits Definition A circuit is ℓ -local if each output bit depends on at most ℓ input bits. January 30, 2019 15 / 51

Parity Halving Problem Locality in circuits Definition A circuit is ℓ -local if each output bit depends on at most ℓ input bits. Fact A strategy for the game implies a 1 -local circuit for PHP . January 30, 2019 15 / 51

Parity Halving Problem Locality in circuits Definition A circuit is ℓ -local if each output bit depends on at most ℓ input bits. Fact A strategy for the game implies a 1 -local circuit for PHP . Can improve game hardness to 1-local hardness. Theorem (1-local hardness) A 1 -local classical circuit solves PHP n on a random input w.p. ≤ 1 2 + 2 −⌈ n / 2 ⌉ . January 30, 2019 15 / 51

Parity Halving Problem Locality ℓ > 1 Idea Reduce to 1-local circuit January 30, 2019 16 / 51

Parity Halving Problem Locality ℓ > 1 Idea Reduce to 1-local circuit by restricting some input bits. January 30, 2019 16 / 51

Parity Halving Problem Locality ℓ > 1 Idea Reduce to 1-local circuit by restricting some input bits. How do we reduce locality to 1? What problem does a circuit for PHP solve after restriction? January 30, 2019 16 / 51

Parity Halving Problem Locality ℓ > 1 Lemma Consider a circuit with n inputs, n outputs, and locality ℓ . We can find a subset of Ω( n ℓ 2 ) input bits such that restricting all other inputs gives a 1 -local circuit. January 30, 2019 17 / 51

Parity Halving Problem Locality ℓ > 1 Lemma Consider a circuit with n inputs, n outputs, and locality ℓ . We can find a subset of Ω( n ℓ 2 ) input bits such that restricting all other inputs gives a 1 -local circuit. Proof. Consider a graph with a vertex for each input bit, an edge if both inputs affect some common output bit. January 30, 2019 17 / 51

Parity Halving Problem Locality ℓ > 1 Lemma Consider a circuit with n inputs, n outputs, and locality ℓ . We can find a subset of Ω( n ℓ 2 ) input bits such that restricting all other inputs gives a 1 -local circuit. Proof. Consider a graph with a vertex for each input bit, an edge if both inputs affect some common output bit. Choose an independent set of vertices to get locality 1. January 30, 2019 17 / 51

Parity Halving Problem Locality ℓ > 1 Lemma Consider a circuit with n inputs, n outputs, and locality ℓ . We can find a subset of Ω( n ℓ 2 ) input bits such that restricting all other inputs gives a 1 -local circuit. Proof. Consider a graph with a vertex for each input bit, an edge if both inputs affect some common output bit. Choose an independent set of vertices to get locality 1. an’s theorem: Largest independent set has size Ω( n 2 / | E | ). Tur´ January 30, 2019 17 / 51

Parity Halving Problem Locality ℓ > 1 Lemma Consider a circuit with n inputs, n outputs, and locality ℓ . We can find a subset of Ω( n ℓ 2 ) input bits such that restricting all other inputs gives a 1 -local circuit. Proof. Consider a graph with a vertex for each input bit, an edge if both inputs affect some common output bit. Choose an independent set of vertices to get locality 1. an’s theorem: Largest independent set has size Ω( n 2 / | E | ). Tur´ Each output is responsible for at most O ( ℓ 2 ) edges = ⇒ | E | = O ( n ℓ 2 ). January 30, 2019 17 / 51

Parity Halving Problem Outputs Inputs January 30, 2019 18 / 51

Parity Halving Problem Restrictions of PHP What happens when we take a circuit for PHP and fix some input bits? January 30, 2019 19 / 51

Parity Halving Problem Restrictions of PHP What happens when we take a circuit for PHP and fix some input bits? E.g., x n = 1 = ⇒ remaining inputs have odd parity. January 30, 2019 19 / 51

Parity Halving Problem Restrictions of PHP What happens when we take a circuit for PHP and fix some input bits? E.g., x n = 1 = ⇒ remaining inputs have odd parity. Parity Halving Problem (Original) Given x ∈ { 0 , 1 } n such that | x | ≡ 0 (mod 2), output y ∈ { 0 , 1 } n such that | x | ≡ 0 (mod 4) = ⇒ | y | ≡ 0 (mod 2) , | x | ≡ 2 (mod 4) = ⇒ | y | ≡ 1 (mod 2) . January 30, 2019 19 / 51

Parity Halving Problem Restrictions of PHP What happens when we take a circuit for PHP and fix some input bits? E.g., x n = 1 = ⇒ remaining inputs have odd parity. Parity Halving Problem (Variant 1) Given x ∈ { 0 , 1 } n such that | x | ≡ 1 (mod 2), output y ∈ { 0 , 1 } n such that | x | ≡ 3 (mod 4) = ⇒ | y | ≡ 0 (mod 2) , | x | ≡ 1 (mod 4) = ⇒ | y | ≡ 1 (mod 2) . January 30, 2019 19 / 51

Parity Halving Problem Restrictions of PHP What happens when we take a circuit for PHP and fix some input bits? E.g., x n = 1 = ⇒ remaining inputs have odd parity. Parity Halving Problem (Variant 2) Given x ∈ { 0 , 1 } n such that | x | ≡ 0 (mod 2), output y ∈ { 0 , 1 } n such that | x | ≡ 2 (mod 4) = ⇒ | y | ≡ 0 (mod 2) , | x | ≡ 0 (mod 4) = ⇒ | y | ≡ 1 (mod 2) . January 30, 2019 19 / 51

Parity Halving Problem Restrictions of PHP What happens when we take a circuit for PHP and fix some input bits? E.g., x n = 1 = ⇒ remaining inputs have odd parity. Parity Halving Problem (Variant 3) Given x ∈ { 0 , 1 } n such that | x | ≡ 1 (mod 2), output y ∈ { 0 , 1 } n such that | x | ≡ 1 (mod 4) = ⇒ | y | ≡ 0 (mod 2) , | x | ≡ 3 (mod 4) = ⇒ | y | ≡ 1 (mod 2) . January 30, 2019 19 / 51

Parity Halving Problem Restrictions of PHP What happens when we take a circuit for PHP and fix some input bits? E.g., x n = 1 = ⇒ remaining inputs have odd parity. Parity Halving Problem (All Variants) Given x ∈ { 0 , 1 } n such that | x | ≡ b (mod 2), output y ∈ { 0 , 1 } n such that | x | ≡ b (mod 4) = ⇒ | y | ≡ 0 (mod 2) , | x | ≡ b + 2 (mod 4) = ⇒ | y | ≡ 1 (mod 2) . where b ∈ { 0 , 1 , 2 , 3 } . January 30, 2019 19 / 51

Parity Halving Problem Restrictions of PHP What happens when we take a circuit for PHP and fix some input bits? E.g., x n = 1 = ⇒ remaining inputs have odd parity. Parity Halving Problem (All Variants) Given x ∈ { 0 , 1 } n such that | x | ≡ b (mod 2), output y ∈ { 0 , 1 } n such that | x | ≡ b (mod 4) = ⇒ | y | ≡ 0 (mod 2) , | x | ≡ b + 2 (mod 4) = ⇒ | y | ≡ 1 (mod 2) . where b ∈ { 0 , 1 , 2 , 3 } . Claim All problems are equivalent. January 30, 2019 19 / 51

Parity Halving Problem Locality- ℓ Hardness Result Theorem 2 + 2 − Ω( n /ℓ 2 ) . An ℓ -local classical circuit solves PHP n on a random input w.p. ≤ 1 Proof. Find Ω( n /ℓ 2 ) inputs with non-overlapping light cones. Fix the rest. The remaining circuit solves a variant of PHP on Ω( n /ℓ 2 ) bits. January 30, 2019 20 / 51

Parity Halving Problem Locality- ℓ Hardness Result Theorem 2 + 2 − Ω( n /ℓ 2 ) . An ℓ -local classical circuit solves PHP n on a random input w.p. ≤ 1 Proof. Find Ω( n /ℓ 2 ) inputs with non-overlapping light cones. Fix the rest. The remaining circuit solves a variant of PHP on Ω( n /ℓ 2 ) bits. Corollary Since NC 0 circuits have locality ℓ = O (1) , they solve PHP w.p. 1 2 + 2 − Ω( n ) . January 30, 2019 20 / 51

Parity Halving Problem AC 0 hardness Problem Unbounded fan-in gates make it easy to have locality n . January 30, 2019 21 / 51

Parity Halving Problem AC 0 hardness Problem Unbounded fan-in gates make it easy to have locality n . Solution Finally some circuit complexity theory: the Switching Lemma!! January 30, 2019 21 / 51

Parity Halving Problem AC 0 hardness Problem Unbounded fan-in gates make it easy to have locality n . Solution Finally some circuit complexity theory: the Switching Lemma!! Switching Lemma (Intuition) Consider an AC 0 circuit. With high probability, restricting a (large) random subset of bits produces a circuit with n o (1) locality. January 30, 2019 21 / 51

Parity Halving Problem Switching Lemma in Action Theorem AC 0 circuits solve PHP w.p. at most 1 2 + o (1) . Proof. Apply the switching lemma. Locality is reduced, and the resulting circuit solves a variant of PHP, so hardness for local circuits implies 1 2 + o (1) probability of success. January 30, 2019 22 / 51

Relaxed Parity Halving Problem Section 3 Relaxed Parity Halving Problem January 30, 2019 23 / 51

Relaxed Parity Halving Problem We don’t want to use an advice state, but we can’t construct it ourselves. Theorem 1 2 ( | 0 n � + | 1 n � ) cannot be constructed in QNC 0 . The state √ January 30, 2019 24 / 51

Relaxed Parity Halving Problem We don’t want to use an advice state, but we can’t construct it ourselves. Theorem 1 2 ( | 0 n � + | 1 n � ) cannot be constructed in QNC 0 . The state √ But we can construct a poor man’s cat state ! 1 √ ( | z � + | z � ) 2 January 30, 2019 24 / 51

Relaxed Parity Halving Problem We don’t want to use an advice state, but we can’t construct it ourselves. Theorem 1 2 ( | 0 n � + | 1 n � ) cannot be constructed in QNC 0 . The state √ But we can construct a poor man’s cat state ! 1 ( | z � + | z � ) = X z 1 ( | 0 n � + | 1 n � ) √ √ 2 2 January 30, 2019 24 / 51

Relaxed Parity Halving Problem We don’t want to use an advice state, but we can’t construct it ourselves. Theorem 1 2 ( | 0 n � + | 1 n � ) cannot be constructed in QNC 0 . The state √ But we can construct a poor man’s cat state ! 1 ( | z � + | z � ) = X z 1 ( | 0 n � + | 1 n � ) √ √ 2 2 A cat state with some bits flipped. January 30, 2019 24 / 51

Relaxed Parity Halving Problem Poor Man’s Cat State Q : If we can construct | z � + | z � = X z | � in QNC 0 , then why can’t we apply X z to get | � ? January 30, 2019 25 / 51

Relaxed Parity Halving Problem Poor Man’s Cat State Q : If we can construct | z � + | z � = X z | � in QNC 0 , then why can’t we apply X z to get | � ? A : We don’t know what z is!! January 30, 2019 25 / 51

Relaxed Parity Halving Problem Poor Man’s Cat State Q : If we can construct | z � + | z � = X z | � in QNC 0 , then why can’t we apply X z to get | � ? A : We don’t know what z is!! Theorem In QNC 0 we can 1 2 ( | z � + | z � ) for some uniformly random z ∈ { 0 , 1 } n , construct √ with information d ∈ { 0 , 1 } n − 1 from which z can be recovered (up to complement) in AC 0 . January 30, 2019 25 / 51

Relaxed Parity Halving Problem Poor Man’s Cat State Example Consider a tree G = ( V , E ). Let there be a | + � qubit for each vertex. z 1 z 4 z 5 z 2 z 3 January 30, 2019 26 / 51

Relaxed Parity Halving Problem Poor Man’s Cat State Example For each edge, measure the parity of the two endpoints. z 1 z 4 z 1 ⊕ z 5 = 1 z 4 ⊕ z 5 = 1 z 5 z 2 ⊕ z 5 = 0 z 3 ⊕ z 5 = 1 z 2 z 3 January 30, 2019 27 / 51

Relaxed Parity Halving Problem Poor Man’s Cat State Example Two vectors, z and z , are consistent with these measurements. 0 0 0 ⊕ 1 = 1 0 ⊕ 1 = 1 1 1 ⊕ 1 = 0 0 ⊕ 1 = 1 1 0 January 30, 2019 28 / 51

Relaxed Parity Halving Problem Poor Man’s Cat State Example To construct z , let z i be the parity of the path from z 1 to z i . z 1 = 0 1 1 0 1 z i = 1 January 30, 2019 29 / 51

Relaxed Parity Halving Problem Poor Man’s Cat State Example To construct z , let z i be the parity of the path from z 1 to z i . z 1 = 0 1 1 0 1 z i = 1 z 2 = z 1 ⊕ z 2 = ( z 1 ⊕ z 5 ) ⊕ ( z 2 ⊕ z 5 ) = 1 ⊕ 0 = 1 January 30, 2019 29 / 51

Relaxed Parity Halving Problem Poor Man’s Cat State Example Final output is state | 01001 � + | 10110 � (vertex qubits) and d = 1011 (edge measurements). 0 0 1 1 1 0 1 1 0 January 30, 2019 30 / 51

Relaxed Parity Halving Problem What kind of tree to use? Line graph! January 30, 2019 31 / 51

Relaxed Parity Halving Problem What kind of tree to use? Line graph! We want low diameter, so it is easier to compute z from d . January 30, 2019 31 / 51

Relaxed Parity Halving Problem What kind of tree to use? Line graph! We want low diameter, so it is easier to compute z from d . Star graph! January 30, 2019 31 / 51

Relaxed Parity Halving Problem What kind of tree to use? Line graph! We want low diameter, so it is easier to compute z from d . Star graph! We want low degree, since edges incident at the same vertex cannot be simultaneously measured. January 30, 2019 31 / 51

Relaxed Parity Halving Problem What kind of tree to use? Line graph! We want low diameter, so it is easier to compute z from d . Star graph! We want low degree, since edges incident at the same vertex cannot be simultaneously measured. Balanced binary tree! Max degree ∆ = 3, diameter d = Θ(log n ). (AC 0 can compute O (log n ) size parities) January 30, 2019 31 / 51

Relaxed Parity Halving Problem Relaxed Parity Halving Problem Q: What do we do with the poor man’s cat state? January 30, 2019 32 / 51

Relaxed Parity Halving Problem Relaxed Parity Halving Problem Q: What do we do with the poor man’s cat state? A: Pretend it’s a cat state and run the same algorithm!! January 30, 2019 32 / 51

Relaxed Parity Halving Problem Relaxed Parity Halving Problem Q: What do we do with the poor man’s cat state? A: Pretend it’s a cat state and run the same algorithm!! January 30, 2019 32 / 51

Relaxed Parity Halving Problem x 1 x 1 x 2 x 2 x 3 x 3 y 1 X S H y 2 | � S H y 3 X S H January 30, 2019 33 / 51

Relaxed Parity Halving Problem x 1 x 1 x 2 x 2 x 3 x 3 y 1 S Z X H y 2 | � S H y 3 S Z X H January 30, 2019 34 / 51

Relaxed Parity Halving Problem x 1 x 1 x 2 x 2 x 3 x 3 y 1 S Z H Z y 2 | � S H y 3 S Z H Z January 30, 2019 35 / 51

Relaxed Parity Halving Problem x 1 x 1 x 2 x 2 x 3 x 3 y 1 S Z H y 2 | � S H y 3 S Z H January 30, 2019 36 / 51

Relaxed Parity Halving Problem x 1 x 1 x 2 x 2 x 3 x 3 y 1 S H X y 2 | � S H y 3 S H X January 30, 2019 37 / 51

Relaxed Parity Halving Problem Relaxed Parity Halving Problem Given an even parity input x ∈ { 0 , 1 } n , output y ∈ { 0 , 1 } n such that | y | ≡ 1 2 | x | + � x , z � (mod 2) . January 30, 2019 38 / 51

Relaxed Parity Halving Problem Relaxed Parity Halving Problem Given an even parity input x ∈ { 0 , 1 } n , output y ∈ { 0 , 1 } n and d ∈ { 0 , 1 } n − 1 such that | y | ≡ 1 2 | x | + � x , z � (mod 2) where z ∈ { 0 , 1 } n is either vector consistent d . January 30, 2019 38 / 51

Relaxed Parity Halving Problem Relaxed Parity Halving Problem Given an even parity input x ∈ { 0 , 1 } n , output y ∈ { 0 , 1 } n and d ∈ { 0 , 1 } n − 1 such that | y | ≡ 1 2 | x | + � x , z � (mod 2) where z ∈ { 0 , 1 } n is either vector consistent d . RPHP ∈ QNC 0 is clear. January 30, 2019 38 / 51