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Eliminating Intermediate Measurements in Space-Bounded Quantum Computation
Bill Fefferman and Zack Remscrim (University of Chicago)
+
Quantum Logspace Algorithm for Powering Matrices with Bounded Norm
Uma Girish, Ran Raz and Wei Zhan (Princeton University)
Principle of Deferred Measurement
H H|0β© H H|0β© = 0, always
= α0, Prob 1
2
1, Prob 12
Okay if only care about time
Not okay if also care about spaceCost = 1 ancilla (space) per measurementEx: log(π)-space and ππππ¦(π)-time
May have ππππ¦(π) measurementsβ exponential blowup to ππππ¦(π)-space
= α0, Prob 1
2
1, Prob 12
Possible without space blowup? [Watrousβ01, Watrousβ03, van Melkebeek-Watsonβ12, Ta-Shmaβ13, Fefferman-Linβ16, etc.]
= α0, Prob 1
2
1, Prob 12
|0β©
H H|0β©
Simulate measurement
Main Results
Can eliminate intermediate measurement without space blowup or time blowup
Quantum logspace algorithms and matching hardness results for many natural linear algebraic problems For well-conditioned matrix π΄, approximate
det(π΄)π΄β1
π΄π
etc.β can eliminate intermediate measurements
Why is Space Important?
Current experimental quantum computers: Noisy Intermediate Scale era [Preskillβ18]Intermediate Scale = few qubits
IBM Q System53 QubitsCost: β« $400
Apple iPhone 7128 GBCost: $400
(somewhat unfair comparison)
Near-term quantum computers have few qubits β space is important
Why Eliminate Intermediate Measurements?
Measurements are a natural resource, just like time and space
Unitary computations (i.e., without intermediate measurements) are reversibleUndo computation: useful in design/analysis of algorithms
βTidyβ subroutine calling [Bennet-Bernstein-Brassard-Vaziraniβ97]Quantum rewinding [Watrousβ09]
No heat generation (Landauerβs Principle)
Shows definition of βquantum space π π β is robustgeneral = unitaryFor probabilistic space
β general quantum (easy)β unitary quantum (previously unknown)
Our Model of Space Bounded Computation
We require that there is a classical deterministic space πΊ Turing Machine which on input π₯ β 0,1 β, outputs the description of the update operations.
An algorithm computes a function π: 0,1 π β 0,1 π if the output of the algorithm is π(π₯)
(with probability at least 2
3if the algorithm is randomized/quantum).
0
0
0
0
0
0
1
0
1
1
1
1
1
0
0
0
0
0
0
0
Update
memory
Update
memory
β¦
Time π β€ 2π(π)
Spac
e π
Ou
tpu
t
Logspace Algorithms:Space π(log π) and Time poly(π)
Classical Logspace Algorithms:
π³: Classical algorithms where the updates are deterministic transition matrices.π©π·π³ : Classical randomized algorithms where the updates are stochastic matrices.
Quantum Logspace Algorithms:
π©πΈπΌπ³ : (Unitary Quantum algorithms)β’ Store qubits.β’ Updates are by unitary quantum channels.β’ The output is the outcome of measuring some qubits at the end of the computation.
π©πΈπΈπ³ : (Pure Quantum algorithms)
β’ Updates are by unital quantum channels.π©πΈπ³ : (Quantum algorithms)
β’ Updates are by general quantum channels.
Logspace Algorithms:Space π(log π) and Time poly(π)
π©πΈπΌπ³Unitary Logspace
π©πΈπΈπ³
Pure Quantum Logspace
π©πΈπ³General Quantum Logspace
Unitary quantum channels Unital quantum channels General quantum channels
β Unitary operations. β Unitary operations.β Intermediate Measurements.
β Unitary operations.β Intermediate Measurements.
β Reset qubits.
Purely quantum memory,Unitary Operations
Purely quantum memory,Unitary Operations
+ Intermediate Measurements
Quantum Memory,Unitary Operations,
Intermediate Measurements+ Classical Memory
+ Copy measurement outcome to classical memory
Our Result:
Contraction Matrix Powering is in π©πΈπΌπ³.
Given: An π Γ π real contraction matrix π΄ (i.e., π΄ β€ 1),
π β€ poly(π), π β₯1
poly(π)and indices π , π‘ β 1, β¦ , π
Estimate AT[π , π‘] up to π additive error.
We [Girish-Raz-Zhanβ20] show that this problem can be solved in π©πΈπΌπ³. Equivalently, Iterated Contraction Matrix Multiplication is in π©πΈπΌπ³.
It is not known if this problem can be solved in π©π·π³.
Applications:
Eliminating Intermediate MeasurementsWe [Girish-Raz-Zhanβ20] show that π©πΈπΌπ³ = π©πΈπΈπ³.
Measurements during computation donβt give more power to quantum logspace algorithms with only quantum memory.
In general, we show that a pure quantum algorithm of space π and time π with intermediate measurements,
can be simulated in space π(π + logπ) and time poly(π, 2π(π)) without intermediate measurements.
β π©πΈπΌπΊπ·π¨πͺπ¬ π π = π©πΈπΈπΊπ·π¨πͺπ¬(π π ) for any space-constructible π π = Ξ©(log π ).
[Fefferman-Remscrimβ20] show that π©πΈπΌπ³ = π©πΈπΈπ³ = π©πΈπ³.
β π©πΈπΌπΊπ·π¨πͺπ¬ π π = π©πΈπΊπ·π¨πͺπ¬(π π ) for any space-constructible π π = Ξ©(log π ).
The ability to do intermediate measurements or reset qubits doesnβt give more power to quantum logspace algorithms.
Applications:
Derandomizing π©πΈπΈπ³
π¨ π© denotes that π© is at least as powerful as π¨.
We show that π©πΈπΈπ³ algorithms donβt require randomness.
π³
π©πΈπΌπ³
π©π·π³
π©πΈπΈπ³ π©πΈπ³
+ randomness
+ randomness
+ reset qubits
[Girish-Raz-Zhanβ20]
[Fefferman-Remscrimβ20]
Applications:
π©π·π³ versus π©πΈπΌπ³
π¨ π© denotes that π© is at least as powerful as π¨.
π³
π©πΈπΌπ³
π©π·π³
π©πΈπΈπ³ π©πΈπ³
+ randomness
+ randomness
+ reset qubits
[Girish-Raz-Zhanβ20]
[Fefferman-Remscrimβ20]
?
?
Connection between Contraction Matrix Multiplication and π©πΈπΈπ³
|0β©
|0β©
πΌπ πΌππΌπ»
β¦
β¦
A generic π©πΈπΈπ³ algorithm
The probability that this circuit outputs 1 is precisely
vec(π·ππ·πβ )β πΌπ» βπΌπ» π΄β¦ πΌπ βπΌπ π΄ πΌπ βπΌπ (ππ βππ)
Ξ 1 = projection matrix onto πβ© βΆ π1 = 1 }π£0 = 0π
π = a diagonal matrix with diagonal entries in 0,1 .
Unitary matrices
Contraction matrices Unit vectorVectorβ
= πβ (π¨poly(π)Γβ―Γ π¨π)π
Our ApproachStep 1. Embed each of the contraction matrices π¨π, β¦ , π¨π» inside unitary logspace quantum circuits.
π΄ βͺπ΄ π β π΄π΄β
π β π΄β π΄ βπ΄β
Step 2. Compose the quantum circuits carefully.
π΄ β 0β β 00 0 π
Γπ΅ 0 β0 π 0β 0 β
=π΄π΅ β ββ β ββ β β
Step 3. Estimate πβ π΄π using a quantum circuit that embeds π΄.
An argument similar to Groverβs search.
Well-Conditioned Matrix Inversion
Given: invertible π Γ π matrix π΄ and indices π, π β 1, β¦ , πApprox π΄β1[π, π] to precision 1/ππππ¦ π
Difficulty depends on condition number π π΄ =π1(π΄)
ππ(π΄)
π1 π΄ = largest singular value, ππ π΄ = smallest singular value
Well-conditioned: π π΄ = ππππ¦ π , Different params (sparse 2π Γ 2π matrix) in π©πΈπ· [Harrow-Hassidim-Lloydβ09]in π©πΈπ³ [Ta-Shmaβ13] is π©πΈπΌπ³-complete [Fefferman-Linβ16]
Determinant
π«π¬π» = problems reducible to computing det(π΄) [Cookβ85] π Γ π matrix π΄, entries are π-bit integers
Natural complete problems: Determinant, Matrix Inversion, Matrix Powering, Iterated Matrix Multiplication, etc.
ππππ¦-conditioned matrix inversion is π©πΈπΌπ³-complete [Fefferman-Linβ16]
We generalize: ππππ¦-conditioned π«π¬π»-complete problems are π©πΈπΌπ³-completeβ can eliminate intermediate measurements
Difficulty: βstandardβ reductions generally do not preserve being well-conditioned
Well-Conditioned Matrix PoweringGiven: π Γ π matrix π΄, π‘ β€ ππππ¦(π), and indices π, π β 1, β¦ , π
Approx π΄π‘[π, π] to precision 1/ππππ¦ π
Promise: π΄π β€ ππππ¦ π , βπ β 1,β¦ , π‘ (analogue of βpoly-conditionedβ for powering)
Simple reduction to ππππ¦-conditioned matrix inversion ββ π©πΈπΌπ³
πΌ βπ΄ 0 0 0 0
0 πΌ βπ΄ 0 0 0
0 0 πΌ βπ΄ 0 0
0 0 0 πΌ βπ΄ 0
0 0 0 0 πΌ βπ΄
0 0 0 0 0 πΌ
π΅ =
πΌ π΄ π΄2 π΄3 π΄4 π΄5
0 πΌ π΄ π΄2 π΄3 π΄4
0 0 πΌ π΄ π΄2 π΄3
0 0 0 πΌ π΄ π΄2
0 0 0 0 πΌ π΄
0 0 0 0 0 πΌ
π΅β1 =
π΄π‘ appears in top-right block of π΅β1 and π΅ is ππππ¦-conditioned
Well-Conditioned Iterated Matrix Product
Similar to powering: Now approx entry of π΄1β―π΄π‘, π‘ β€ ππππ¦(π), π΄πβ―π΄β β€ ππππ¦ π , β 1 β€ π β€ β β€ π‘
Can reduce to ππππ¦-conditioned matrix inversion ββ π©πΈπΌπ³
Is π©πΈπ³-hard: General quantum circuit is sequence of (arbitrary) quantum channels
β π©πΈπ³ = π©πΈπΌπ³
β π©πΈπΊπ·π¨πͺπ¬ π π = π©πΈπΌπΊπ·π¨πͺπ¬(π π ), any space-constructible π π = Ξ©(log π )
Also show results for πΉπΈπ³ and π΅πΈπ³, and for βπΈπ΄π¨β and βπ«πΈπͺπβ versions of π©πΈπ³,πΉπΈπ³, and π΅πΈπ³
Ξ¦1
|0β©
|0β©
|0β©
Ξ¦2 Ξ¦π‘β― πππ’π‘
Well-Conditioned Determinant
Given: π Γ π matrix π΄, π π΄ = ππππ¦(π)Approx log( det π΄ ) to precision 1/ππππ¦ π (Approx det π΄ to multiplicative factor 1 + 1/ππππ¦ π )
We show: is π©πΈπ³(= π©πΈπΌπ³)-completeOther well-conditioned π«π¬π»-complete also π©πΈπ³-complete βStandardβ reductions do not preserve well-conditioned (e.g. Berkowitzβs algorithm)Our reductions: various power series approximations, quantum βinstance compressionβ
[Boix-Adsera, Eldar, and Mehrabanβ19]: β π«πΊπ·π¨πͺπ¬(log2 π ππππ¦(log log π))Used (different) power series approximations
Suggests source of quantum advantageWe show β π«πΊπ·π¨πͺπ¬(log2 π)
Recall: π©πΈπ³ β π«πΊπ·π¨πͺπ¬(log2 π) [Watrousβ03] Improve dependence on π π΄ ?
Would show π©πΈπ³ β π«πΊπ·π¨πͺπ¬(log2βπΏ π)