![Page 1: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/1.jpg)
Limitations on transversal gates
Michael Newman1 and Yaoyun Shi1,2
University of Michigan1, Alibaba Group2
February 6, 2018
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Quantum fault-tolerance
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Quantum fault-tolerance
![Page 4: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/4.jpg)
Quantum fault-tolerance
![Page 5: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/5.jpg)
Quantum fault-tolerance
![Page 6: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/6.jpg)
What are transversal gates?
![Page 7: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/7.jpg)
What are transversal gates?
![Page 8: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/8.jpg)
What are transversal gates?
![Page 9: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/9.jpg)
What are transversal gates?
![Page 10: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/10.jpg)
What are transversal gates?
![Page 11: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/11.jpg)
What are transversal gates?
![Page 12: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/12.jpg)
What are transversal gates?
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What are transversal gates?
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What are transversal gates?
Definition
A transversal p-qubit gate on an n-qubit codespace C is a logical gateU : U(C⊗p) = C⊗p that decomposes as
U =n⊗
j=1
Uj
where each Uj acts on a single subsystem of the code.
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What are transversal gates?
Logical gates implemented transversally are called bd−12 c-fault-tolerant: if
at most bd−12 c components fail during their application, the resultingerrors can be corrected.
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What are transversal gates?
![Page 17: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/17.jpg)
What are transversal gates?
![Page 18: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/18.jpg)
What are transversal gates?
![Page 19: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/19.jpg)
What are transversal gates?
![Page 20: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/20.jpg)
What are transversal gates?
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No quantum universal transversal gate set
Theorem (Eastin-Knill ’09)
For any fixed quantum code that can detect at least 1 error, the set oflogically distinct transversal gates must form a finite group, and so cannotbe universal for quantum computing.
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No quantum universal transversal gate set
Proof.
A code C is error-detecting if and only if PEP ∝ P for all |E | = 1.
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Importance of a fixed partition
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Importance of a fixed partition
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Importance of a fixed partition
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Importance of a fixed partition
![Page 27: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/27.jpg)
Importance of a fixed partition
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Universal gate sets
〈H,P,CX 〉 transversally (e.g. 2D-color codes); 〈T 〉 magic-statedistillation.
〈CCZ 〉 transversally; 〈H〉 gauge-fixing or state-preparation +gate-teleportation (e.g. 3D-color codes, 3D-surface codes).
Theorem (Shi ’03)
Toffoli and Hadamard together form a universal gate set for quantumcomputing.
Definition
Toffoli: |a〉|b〉|c〉 7→ |a〉|b〉|c ⊕ a · b〉, and is universal for classical reversiblecomputing.
![Page 29: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/29.jpg)
Universal gate sets
〈H,P,CX 〉 transversally (e.g. 2D-color codes); 〈T 〉 magic-statedistillation.
〈CCZ 〉 transversally; 〈H〉 gauge-fixing or state-preparation +gate-teleportation (e.g. 3D-color codes, 3D-surface codes).
Theorem (Shi ’03)
Toffoli and Hadamard together form a universal gate set for quantumcomputing.
Definition
Toffoli: |a〉|b〉|c〉 7→ |a〉|b〉|c ⊕ a · b〉, and is universal for classical reversiblecomputing.
![Page 30: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/30.jpg)
Universal gate sets
〈H,P,CX 〉 transversally (e.g. 2D-color codes); 〈T 〉 magic-statedistillation.
〈CCZ 〉 transversally; 〈H〉 gauge-fixing or state-preparation +gate-teleportation (e.g. 3D-color codes, 3D-surface codes).
Theorem (Shi ’03)
Toffoli and Hadamard together form a universal gate set for quantumcomputing.
Definition
Toffoli: |a〉|b〉|c〉 7→ |a〉|b〉|c ⊕ a · b〉, and is universal for classical reversiblecomputing.
![Page 31: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/31.jpg)
A question
Toffoli gates form the dominating portion of many useful quantumalgorithms (e.g. Shor’s algorithm).
Question: Can we implement the Toffoli gate transversally on anyquantum error-correcting code?
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A question
Toffoli gates form the dominating portion of many useful quantumalgorithms (e.g. Shor’s algorithm).
Question: Can we implement the Toffoli gate transversally on anyquantum error-correcting code?
![Page 33: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/33.jpg)
A link to cryptography
(Transversal) logical gates ↔ (locally) compute on encoded data
Homomorphic circuits ↔ compute on encrypted data
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A link to cryptography
(Transversal) logical gates ↔ (locally) compute on encoded data
Homomorphic circuits ↔ compute on encrypted data
![Page 35: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/35.jpg)
(Quantum) Homomorphic Encryption
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(Quantum) Homomorphic Encryption
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(Quantum) Homomorphic Encryption
![Page 38: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/38.jpg)
(Quantum) Homomorphic Encryption
![Page 39: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/39.jpg)
(Quantum) Homomorphic Encryption
![Page 40: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/40.jpg)
(Quantum) Homomorphic Encryption
![Page 41: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/41.jpg)
(Quantum) Homomorphic Encryption
We call such a scheme fully-homomorphic if the set of allowableevaluations can be any Boolean circuit.
We call such a scheme compact and non-leveled if the Alice’s work iscompletely independent of the circuit being evaluated.
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(Quantum) Homomorphic Encryption
We call such a scheme fully-homomorphic if the set of allowableevaluations can be any Boolean circuit.
We call such a scheme compact and non-leveled if the Alice’s work iscompletely independent of the circuit being evaluated.
![Page 43: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/43.jpg)
Information-theoretically secure QHE (OTF’15)
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Information-theoretically secure QHE
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Information-theoretically secure QHE
Logical Clifford gates 〈H,P,CX 〉 are transversal on this randomizedencoding.
![Page 46: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/46.jpg)
Information-theoretically secure QHE
Logical Clifford gates 〈H,P,CX 〉 are transversal on this randomizedencoding.
![Page 47: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/47.jpg)
Information-theoretically secure QHE
Logical Clifford gates 〈H,P,CX 〉 are transversal on this randomizedencoding.
![Page 48: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/48.jpg)
Information-theoretically secure QHE
Logical Clifford gates 〈H,P,CX 〉 are transversal on this randomizedencoding.
![Page 49: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/49.jpg)
Information-theoretically secure QHE
Logical Clifford gates 〈H,P,CX 〉 are transversal on this randomizedencoding.
![Page 50: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/50.jpg)
Information-theoretically secure QHE
Let m be the number of noisy qubits that we embed into, and let n be thesize of the code. Then for any two p-qubit input ρ, σ with encryptions ρ, σ,
‖ρ− σ‖1 ≤ 2(4p − 1)
(n + m
n
)− 12
.
Information-theoretic security for encoding sizes scaling polynomially in theinput length.
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Information-theoretically secure QHE
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Information-theoretically secure QHE
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Information-theoretically secure QHE
![Page 54: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/54.jpg)
Information-theoretically secure QHE
![Page 55: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/55.jpg)
Information-theoretically secure QHE
The transversal gates for the code are the homomorphisms for theencryption scheme, as
R (Ed−1 [(U1 ⊗ U2 ⊗ . . .⊗ Un) · |ψ〉L]) = UL|ψ〉L.
Let m be the size of the noise we embed into. Under mild assumptions onthe code, for any p-qubit inputs ρ, σ with encryptions ρ, σ, there exists ac > 0 so that
‖ρ− σ‖1 ≤
((m − 1
m
)n−d+1
− 1 + 2−pc(
2 · 2p
m
)n−d+1) 1
2
.
In order to be secure, require m = 2pc′
for some c ′ < 1 for total encodingsize (n − d + 1)p2pc
′.
![Page 56: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/56.jpg)
Information-theoretically secure QHE
The transversal gates for the code are the homomorphisms for theencryption scheme, as
R (Ed−1 [(U1 ⊗ U2 ⊗ . . .⊗ Un) · |ψ〉L]) = UL|ψ〉L.
Let m be the size of the noise we embed into. Under mild assumptions onthe code, for any p-qubit inputs ρ, σ with encryptions ρ, σ, there exists ac > 0 so that
‖ρ− σ‖1 ≤
((m − 1
m
)n−d+1
− 1 + 2−pc(
2 · 2p
m
)n−d+1) 1
2
.
In order to be secure, require m = 2pc′
for some c ′ < 1 for total encodingsize (n − d + 1)p2pc
′.
![Page 57: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/57.jpg)
Information-theoretically secure QHE
The transversal gates for the code are the homomorphisms for theencryption scheme, as
R (Ed−1 [(U1 ⊗ U2 ⊗ . . .⊗ Un) · |ψ〉L]) = UL|ψ〉L.
Let m be the size of the noise we embed into. Under mild assumptions onthe code, for any p-qubit inputs ρ, σ with encryptions ρ, σ, there exists ac > 0 so that
‖ρ− σ‖1 ≤
((m − 1
m
)n−d+1
− 1 + 2−pc(
2 · 2p
m
)n−d+1) 1
2
.
In order to be secure, require m = 2pc′
for some c ′ < 1 for total encodingsize (n − d + 1)p2pc
′.
![Page 58: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/58.jpg)
Information-theoretically secure QHE
Suppose there existed a quantum error-detecting code implementing theToffoli gate transversally. Then we would obtain a fully-homomorphicencryption scheme with encoding size θ(p2pc
′) for some c ′ < 1.
Theorem (Yu, Perez-Delgado, Fitzsimons ’14)
Suppose a QHE scheme with perfect information-theoretic securityimplements a set of homomorphisms S . Then the size of the encodingmust grow as log2(|S |).
![Page 59: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/59.jpg)
Limitations on ε-ITS-QHE
Definition
An (n,m, p)-quantum random access code is an encoding of n bits b intom qubits ρb along with a set of n POVM’s P0
i ,P1i ni=1 satisfying, for all
b ∈ 0, 1n and i ∈ [n],Tr(Pbi
i ρb) ≥ p.
![Page 60: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/60.jpg)
Limitations on ε-ITS-QHE
Definition
An (n,m, p)-quantum random access code is an encoding of n bits b intom qubits ρb along with a set of n POVM’s P0
i ,P1i ni=1 satisfying, for all
b ∈ 0, 1n and i ∈ [n],Tr(Pbi
i ρb) ≥ p.
![Page 61: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/61.jpg)
Limitations on ε-ITS-QHE
Definition
An (n,m, p)-quantum random access code is an encoding of n bits b intom qubits ρb along with a set of n POVM’s P0
i ,P1i ni=1 satisfying, for all
b ∈ 0, 1n and i ∈ [n],Tr(Pbi
i ρb) ≥ p.
![Page 62: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/62.jpg)
Limitations on ε-ITS-QHE
Definition
An (n,m, p)-quantum random access code is an encoding of n bits b intom qubits ρb along with a set of n POVM’s P0
i ,P1i ni=1 satisfying, for all
b ∈ 0, 1n and i ∈ [n],Tr(Pbi
i ρb) ≥ p.
![Page 63: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/63.jpg)
Limitations on ε-ITS-QHE
Definition
An (n,m, p)-quantum random access code is an encoding of n bits b intom qubits ρb along with a set of n POVM’s P0
i ,P1i ni=1 satisfying, for all
b ∈ 0, 1n and i ∈ [n],Tr(Pbi
i ρb) ≥ p.
![Page 64: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/64.jpg)
Limitations on ε-ITS-QHE
Theorem (Nayak ’99)
For H(·) the binary entropy function, any (n,m, p)-QRAC must satisfy
m ≥ n(1− H(p)).
Ex. Could have been a (16, 5, 0.8)-QRAC but not a (16, 5, 0.9)-QRAC.
![Page 65: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/65.jpg)
Limitations on ε-ITS-QHE
Theorem (Nayak ’99)
For H(·) the binary entropy function, any (n,m, p)-QRAC must satisfy
m ≥ n(1− H(p)).
Ex. Could have been a (16, 5, 0.8)-QRAC but not a (16, 5, 0.9)-QRAC.
![Page 66: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/66.jpg)
Limitations on ε-ITS-QHE
Theorem (N.,Shi)
Consider a QHE scheme that is fully-homomorphic. Let f (p) denote thesize of the evaluated ciphertext as a function of the input size p. If, forsome δ > 0 and any p-qubit input ciphertexts ρ, ρ′,
limp→∞
‖ρ− ρ′‖1 < 1− δ,
then f (p) = Ω(2p).
![Page 67: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/67.jpg)
Limitations on ε-ITS-QHE
Proof sketch.
Let s be the encoding size of the QHE scheme and define a QRAC:
Boolean functions f → f -evaluations of encryptions of 0p
Querying index x ∈ 0, 1p → applying a transformation depending on xto the keyspace and decrypting.
This gives a (2p, s, q(δ))-QRAC for some constant q(δ) > 12 .
By Nayak’s bound, s = Ω(2p).
![Page 68: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/68.jpg)
Limitations on ε-ITS-QHE
Proof sketch.
Let s be the encoding size of the QHE scheme and define a QRAC:
Boolean functions f → f -evaluations of encryptions of 0p
Querying index x ∈ 0, 1p → applying a transformation depending on xto the keyspace and decrypting.
This gives a (2p, s, q(δ))-QRAC for some constant q(δ) > 12 .
By Nayak’s bound, s = Ω(2p).
![Page 69: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/69.jpg)
Limitations on ε-ITS-QHE
Proof sketch.
Let s be the encoding size of the QHE scheme and define a QRAC:
Boolean functions f → f -evaluations of encryptions of 0p
Querying index x ∈ 0, 1p → applying a transformation depending on xto the keyspace and decrypting.
This gives a (2p, s, q(δ))-QRAC for some constant q(δ) > 12 .
By Nayak’s bound, s = Ω(2p).
![Page 70: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/70.jpg)
Limitations on ε-ITS-QHE
Proof sketch.
Let s be the encoding size of the QHE scheme and define a QRAC:
Boolean functions f → f -evaluations of encryptions of 0p
Querying index x ∈ 0, 1p → applying a transformation depending on xto the keyspace and decrypting.
This gives a (2p, s, q(δ))-QRAC for some constant q(δ) > 12 .
By Nayak’s bound, s = Ω(2p).
![Page 71: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/71.jpg)
Limitations on ε-ITS-QHE
Proof sketch.
Let s be the encoding size of the QHE scheme and define a QRAC:
Boolean functions f → f -evaluations of encryptions of 0p
Querying index x ∈ 0, 1p → applying a transformation depending on xto the keyspace and decrypting.
This gives a (2p, s, q(δ))-QRAC for some constant q(δ) > 12 .
By Nayak’s bound, s = Ω(2p).
![Page 72: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/72.jpg)
Limitations on transversal gates
Theorem (N.,Shi)
Almost no quantum error-detecting code can implement Toffoli (or anyclassical-universal) gate set transversally.
Exceptional codes are [[n, k, d ]]-codes that decompose as a d-fold productof “subcodes”.
By Knill-Laflamme, these subcodes must satisfy 〈0|E |0〉 = 〈1|E |1〉 for anydetectable E , but there must exist some E for which 〈0|E |1〉 6= 0.
Ex. Shor’s code: |0〉L = (|000〉+ |111〉)⊗3; |1〉L = (|000〉 − |111〉)⊗3.
![Page 73: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/73.jpg)
Limitations on transversal gates
Theorem (N.,Shi)
Almost no quantum error-detecting code can implement Toffoli (or anyclassical-universal) gate set transversally.
Exceptional codes are [[n, k, d ]]-codes that decompose as a d-fold productof “subcodes”.
By Knill-Laflamme, these subcodes must satisfy 〈0|E |0〉 = 〈1|E |1〉 for anydetectable E , but there must exist some E for which 〈0|E |1〉 6= 0.
Ex. Shor’s code: |0〉L = (|000〉+ |111〉)⊗3; |1〉L = (|000〉 − |111〉)⊗3.
![Page 74: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/74.jpg)
Limitations on transversal gates
Theorem (N.,Shi)
Almost no quantum error-detecting code can implement Toffoli (or anyclassical-universal) gate set transversally.
Exceptional codes are [[n, k, d ]]-codes that decompose as a d-fold productof “subcodes”.
By Knill-Laflamme, these subcodes must satisfy 〈0|E |0〉 = 〈1|E |1〉 for anydetectable E , but there must exist some E for which 〈0|E |1〉 6= 0.
Ex. Shor’s code: |0〉L = (|000〉+ |111〉)⊗3; |1〉L = (|000〉 − |111〉)⊗3.
![Page 75: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/75.jpg)
Limitations on transversal gates
Theorem (N.,Shi)
Almost no quantum error-detecting code can implement Toffoli (or anyclassical-universal) gate set transversally.
Exceptional codes are [[n, k, d ]]-codes that decompose as a d-fold productof “subcodes”.
By Knill-Laflamme, these subcodes must satisfy 〈0|E |0〉 = 〈1|E |1〉 for anydetectable E , but there must exist some E for which 〈0|E |1〉 6= 0.
Ex. Shor’s code: |0〉L = (|000〉+ |111〉)⊗3; |1〉L = (|000〉 − |111〉)⊗3.
![Page 76: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/76.jpg)
Limitations on transversal gates
Corollary (N.,Shi)
No quantum error-detecting stabilizer code can implement the Toffoli gatetransversally. (See also Yoder et al.)
Corollary (N., Shi)
No quantum error-detecting code can implement the Toffoli gate stronglytransversally.
![Page 77: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/77.jpg)
Limitations on transversal gates
Corollary (N.,Shi)
No quantum error-detecting stabilizer code can implement the Toffoli gatetransversally. (See also Yoder et al.)
Corollary (N., Shi)
No quantum error-detecting code can implement the Toffoli gate stronglytransversally.
![Page 78: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/78.jpg)
Conclusion
What we’ve done...
Shown that almost no quantum error-detecting code can implementeven a classical universal gate set transversally.
Improved cryptographic bounds on quantum homomorphic encryption.
In the future...
Close loophole in proof.
Most transversal gate sets look the same. Are they?
Quantum fault-tolerance schemes aimed at classical computing?
![Page 79: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/79.jpg)
Conclusion
What we’ve done...
Shown that almost no quantum error-detecting code can implementeven a classical universal gate set transversally.
Improved cryptographic bounds on quantum homomorphic encryption.
In the future...
Close loophole in proof.
Most transversal gate sets look the same. Are they?
Quantum fault-tolerance schemes aimed at classical computing?
![Page 80: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/80.jpg)
Conclusion
What we’ve done...
Shown that almost no quantum error-detecting code can implementeven a classical universal gate set transversally.
Improved cryptographic bounds on quantum homomorphic encryption.
In the future...
Close loophole in proof.
Most transversal gate sets look the same. Are they?
Quantum fault-tolerance schemes aimed at classical computing?
![Page 81: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/81.jpg)
Conclusion
What we’ve done...
Shown that almost no quantum error-detecting code can implementeven a classical universal gate set transversally.
Improved cryptographic bounds on quantum homomorphic encryption.
In the future...
Close loophole in proof.
Most transversal gate sets look the same. Are they?
Quantum fault-tolerance schemes aimed at classical computing?
![Page 82: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/82.jpg)
Conclusion
What we’ve done...
Shown that almost no quantum error-detecting code can implementeven a classical universal gate set transversally.
Improved cryptographic bounds on quantum homomorphic encryption.
In the future...
Close loophole in proof.
Most transversal gate sets look the same. Are they?
Quantum fault-tolerance schemes aimed at classical computing?
![Page 83: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/83.jpg)
Conclusion
What we’ve done...
Shown that almost no quantum error-detecting code can implementeven a classical universal gate set transversally.
Improved cryptographic bounds on quantum homomorphic encryption.
In the future...
Close loophole in proof.
Most transversal gate sets look the same. Are they?
Quantum fault-tolerance schemes aimed at classical computing?
![Page 84: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/84.jpg)
Conclusion
What we’ve done...
Shown that almost no quantum error-detecting code can implementeven a classical universal gate set transversally.
Improved cryptographic bounds on quantum homomorphic encryption.
In the future...
Close loophole in proof.
Most transversal gate sets look the same. Are they?
Quantum fault-tolerance schemes aimed at classical computing?
![Page 85: Limitations on transversal gatesmgnewman/Duke_Talk.pdfto the keyspace and decrypting. This gives a (2p;s;q( ))-QRAC for some constant q( ) >1 2. By Nayak’s bound, s = (2p). Limitations](https://reader036.vdocuments.site/reader036/viewer/2022071510/612f19211ecc515869433a23/html5/thumbnails/85.jpg)
Hey thanks
Thank you!
arXiv:1704.07798